UNiveitsrnr  OF  CALIFORNIA, 

LCrARTMENTOFCIVILENCSlNCEHIN, 


WORKS   BY 

PROF.  WILLIAM  CAIN 


VAN  NOSTRAND  SCIENCE  SERIES 

Price  50  cents  each. 

No.  3.  Practical  Designing  of  Retaining= 
Walls.  Seventh  Edition,  Thoroughly 
Prevised. 

No.  12.  Theory  of  Voussoir  Arches. 

Fourth  Edition,  Revised  and  Enlarged. 

No.  38.  Maximum  Stresses  in  Framed 
Bridges.  New  and  Revised  Edition. 

No.  42.  Theory  of  Steel  -  Concrete 
Arches,  and  of  Vaulted  Structures. 

Fifth  Edition,  thoroughly  Revised. 

No.  48.  Theory  of  Solid  and  Braced 
Elastic  Arches.  Second  Edition,  Re- 
vised and  Enlarged. 


Brief   Course   in    the   Calculus, 

I2mo.  Cloth.     280  Pages 
Illustrated  Net  $1.75 


PRACTICAL  DESIGNING 


RETAINING  WALLS, 

WITH    APPENDICES    ON 

STRESSES  IN  MASONRY  DAMS 

BT 

PROFESSOR  WILLIAM  CAIN,   A.  M.,  C.  E. 

UNIVERSITY   OP  NOR^H   CAROLINA. 
MEM.   AM.   SOC.    C.  E. 

ILLUSTRATED. 


SEVENTH  EDITION,  THOROUGHLY  REVISED. 


NEW  YORK: 

D.  VAN  NOSTRAND  COMPANY, 

25  PARK  PLACE 

1914 


'?/  V 

Enrincerinj 
Library 


Engineering 
Library 

COPYRIGHT,   1888, 
BY  W.  H.  FARRINGTON. 


COPYRIGHT,   1910, 
BY  D.  VAN  NOSTRAND  COMPANY, 


COPYRIGHT,   1914, 
BY  D.  VAN  NOSTRAND  COMPANY. 


PREFACE   TO  SEVENTH  EDITION 


IN  the  first  six  editions  of  this  work, 
considerable  space  was  given  to  the  results 
of  experiments  on .  model  retaining-walls 
and  rotating  retaining-boards.  As  this 
part  of  the  subject  has  been  fully  discussed 
by  the  writer  in  a  paper  entitled  "  Exper- 
iments on  Retaining-waUs  and  Pressures 
on  Tunnels,"*  it  was  thought  best  to  omit 
a  detailed  discussion  of  the  experiments 
in  this  edition,  particularly  as  an  adequate 
interpretation  requires  the  consideration 
of  the  theory  of  earth  pressure  when  the 
earth  is  supposed  endowed  with  both  fric- 
tion and  cohesion.  More  important  still, 
the  omission  gives  space  for  a  more  ade- 
quate treatment  of  the  designing  of  walls 
of  various  types. 

The    present    work    is    divided    into    an 
Introduction,    where   the   direction   of  the 

*  Transactions  Am.  Soc.  C.E.,  Vol.  LXXII  (1911). 


8O0300 


IV 


earth  thrust  receives  careful  attention,  and 
four  chapters,  pertaining  to  reservoir  walls 
and  the  theory  of  retaining-walls,  developed 
both  by  the  graphical  and  analytical 
methods  and  leading  up,  after  a  short 
discussion  of  experiments,  to  the  practical 
designing  of  retaining  walls. 

The  design  of  five  different  types  of 
retaining-walls  is  given  in  detail  not  only 
for  a  horizontal  earth  surface  but  likewise 
for  the  earth  surface  at  the  angle  of  repose. 

The  tables,  giving  ratio  of  base  to  height, 
for  the  most  familiar  types  of  walls,  should 
prove  especially  valuable  to  constructors. 

In  the  brief  discussion  of  dams,  the 
occasion  is  taken  to  develop  certain  well- 
known  elementary  principles  that  are  com- 
mon to  retaining-walls  as  well  as  dams. 
In  subsequent  chapters  of  this  work  a  good 
deal  of  new  matter  is  given  for  the  first 
time;  notably  in  the  analytical  theory  of 
the  retaining-wall,  and  in  the  graphical 
discussion  of  "  the  limiting  plane  "  in 
Chapter  II.  The  theory  of  the  retaining- 
wall  has  been  deduced,  with  the  one 
assumption  of  a  plane  surface  of  rupture, 
from  well-known  mechanical  laws;  Cou- 


lomb's  "  wedge  of  maximum  thrust " 
being  incidentally  proved  in  the  course  of 
the  demonstration,  but  not  assumed  as  a 
first  principle. 

Appendices  I,  II  and  III  on  Masonry 
Dams,  have  been  added,  leading  to  the 
computation  of  the  "  Stresses  in  a  Masonry 
Dam  "  on  any  plane  not  too  near  the  base. 
The  results,  especially  when  taken  in  con- 
nection with  the  experiments  on  rubber 
dams  made  in  England  by  Messrs.  Wilson 
and  Gore,  are  thought  to  be  of  the  highest 
importance. 

The  limits  of  this  book  preclude  the 
consideration  of  the  stresses  due  to  tem- 
perature changes  and  "  uplift  "  due  to 
water  pressure,  subjects  which  are  now 
engaging  the  serious  attention  of  engineers. 

WM.  CAIN. 

CHAPEL  HILL,  N.  C.,  May  5,  1914. 


TABLE   OF   CONTENTS. 


PAGB 

INTRODUCTION 1 

CHAPTER  I. 
RESERVOIR-WALLS 17 

CHAPTER    II. 

THEORY  OF  RETAINING- WALLS.  —  GRAPH- 
ICAL METHOD 34 

CHAPTER  III. 

THEORY    OF     RETAINING-WALLS.  —  ANA- 
LYTICAL METHOD 78 

CHAPTER   IV. 

EXPERIMENTS.    COMPARISON  WITH  THEORY. 
THE   PRACTICAL    DESIGNING  OF   RETAIN- 

ING-WALLS 114 

vii 


Vlll 

APPENDIX  I. 
DESIGN  FOR  A  VERY  HIGH  MASONRY  DAM.  140 

APPENDIX  II. 

STRESSES  IN  MASONRY  DAMS 149 

APPENDIX  III. 

RELATIONS    BETWEEN    STRESSES    AT  ANY 
POINT  OF  A  DAM. 168 


PRACTICAL  DESIGNING 

OF 

RETAINING-WALLS. 


INTRODUCTION. 

1.  THE  retaining  or  revetment  wall  is 
generally  a  wall  of  masonry,  intended  to 
support  the  pressure  of  a  mass  of  earth  or 
other  material  possessing  some  frietional 
stability.  In  certain  cases,  however,  as  in 
dock-walls,  the  backing  or  filling  —  as  the 
material  behind  the  wall  is  called  —  is  liable 
to  become  in  part  or  wholly  saturated  with 
water,  so  that  the  subject  of  water-pressure 
has  to  be  considered  to  complete  the  inves- 
tigation. In  cases  where  the  filling  is  de- 
posited behind  the  wall  after  it  is  built,  the 
full  pressure  due  to  the  pulverulent  fresh 
earth  or  other  backing  is  experienced ;  and 
the  wall  is  designed  to  meet  such  pressure, 
with  a  certain  factor  of  safety,  as  near  as  \t 


can  be  ascertained.  In  time  the  earth 
becomes  more  or  less  consolidated  by  the 
settling  due  to  gravity,  vibrations,  and  rains, 
from  the  compressibility  of  the  material, 
which  thus  brings  into  action  those  cohesive 
and  chemical  affinities  which  manufacture 
solid  clays  out  of  loosely  aggregated  mate- 
rials, and  often  causes  the  bank  eventually 
even  to  shrink  away  from  the  wall  intended 
to  support  it,  when,  of  course,  there  will  be 
no  pressure  exerted  against  the  wall. 

2.  Where  a  wall  is  built  to  support  the 
face  of  a  cutting,  the  pressure  may  be 
nothing  at  first,  but  it  would  be  very  unwise 
to  make  the  wall  much  thinner  than  in  the 
preceding  case  ;  for  it  is  a  well-known  fact 
of  observation,  that  incessant  rains  often 
saturate  the  ground  of  open  cuttings  to 
such  an  extent  as  to  bring  down  masses  of 
earth,  whose  surface  of  rupture  is  curved, 
being  more  or  less  vertical  at  the  top  and 
approaching  a  cycloid  somewhat  in  section  ; 
the  surface  of  sliding  being  so  lubricated 
by  the  water  that  the  pressure  exerted  hori- 
zontally by  this  sliding  mass  is  even  greater 
than  for  dry  pulverulent  materials.  It  is, 


in  fact,  on  this  account,  as  well  as  from  the 
force  exerted  by  water  in  freezing,  and  from 
the  disturbing  influences  caused  by  the 
passage  of  heavy  trains,  wagons,  etc.,  which 
set  up  vibrations  that  lower  the  co-efficient 
of  friction  of  the  earth,  and  besides  add 
considerably  by  their  weight  to  the  thrust 
of  tie  backing,  that  a  factor  of  safety 
against  overturning  and  sliding  of  the  wall 
is  introduced,  which  factor  in  practice  gen- 
erally varies  between  two  and  three  when 
the  actual  lateral  pressure  of  the  earth  is 
considered. 

3.  It  is  stated  that  retain  ing-walls  in 
Canada  require  a  greater  thickness  at  the 
top  to  resist  the  action  of  frost  than  farther 
south  where  the  frost  does  not  penetrate  the 
ground  to  so  great  a  depth.  Again,  if  the 
strata  in  a  cutting  dip  towards  the  wall, 
with  thin  beds  of  clay,  etc.,  interposed  that 
may  act  as  lubricants  when  wet,  the  press- 
ure against  the  wall  may  become  enormous  ; 
or  if  fresh  earth-filling  is  deposited  upon  an 
inclined  surface  of  rock,  or  other  impervious 
material  that  may  become  slippery  when  the 
water  penetrates  and  accumulates  at  its  sur- 


face,  the  pressure  may  become  much  greater 
than  that  due  to  dry  materials.  It  is  found, 
too,  that  certain  clays  swell  when  exposed 
to  the  air  with  great  force ;  others,  again, 
remain  unchanged.  In  all  such  exceptional 
cases  the  engineer  must  use  his  best  judg- 
ment after  a  careful  study  of  the  material 
he  has  to  deal  with.  The  theory  and 
methods  used  in  this  book  will  not  deal 
with  such  exceptional  cases,  but  simply  with 
dry  or  moist  earth- filling  supported  by  good 
masonry  upon  a  firm  foundation  ;  and  it  is 
believed  the  theory  deduced  will  be  of  mate- 
rial assistance  to  any  one  who  may  have  to 
deal  with  even  very  exceptional  conditions, 
or,  as  in  the  case  of  military  engineers,  with 
the  design  of  revetment-walls  partly  as  a 
means  of  defence. 

4.  When  a  retaining-wall  fails,  it  is  not 
generally  from  not  having  sufficient  section 
for  dry  backing  properly  laid  (in  layers 
horizontal  or  inclined  downwards  from  the 
wall),  but  because  the  earth  has  been  dumped 
in  any  fashion  against  the  wall,  and  no 
4 'weep  holes"  have  been  provided  to  let 
off  the  water  that  is  sure  in  time  of  rains  to 


saturate  the  bank.  If  to  this  is  added  bad 
masonry,  and  a  yielding  foundation,  or  one 
liable  to  be  washed  out,  the  final  destruc- 
tion of  the  wall  can  be  pretty  confidently 
counted  on. 

5.  The  following  little  table  of  weights 
and  angles  of  repose  of  various  materials 
used  in  construction  may  prove  of  assistance, 
but  in  any  actual  case  the  engineer  should 
determine  them  by  actual  experiment :  — 


Weight  per  Cubic 
Foot  in  Pounds. 

Angle  of 
Repose. 

Water   .... 

62.4 

0 

Mud  

102. 

0-? 

Shingle,  gravel 

90-109-120 

35°-48° 

Clay       .     .     . 

120 

140.450 

Gravel  and  earth 

126 

— 

Settled  earth 

120-137 

21°-37° 

Dry  sand   . 

90 

34° 

Damp  sand 

120-128 

350.450- 

Marl  .     .     . 

100 

— 

Brick     .     . 

90-135 

_ 

Mortar  .     . 

86-110 

_ 

Brickwork 

110 

_ 

Masonry    . 

110-144 

_ 

Sandstone  . 

130-157 

_ 

Granite 

164-172 

- 

We  may  assume  generally,  as  safe  values 
for  brickwork,  110  pounds  per  cubic  foot  ; 


and  for  walls,  one-half  ashlar  and  one-half 
rubble  backing,  of  granite  142  pounds,  and 
of  sandstone  120  pounds  per  cubic  foot, 
though  the  last  two  values  are  generally 
exceeded.  For  ordinary  earth  or  sand  filling 
the  angle  of  repose  can  be  taken  at  one  and 
one-half  base  to  one  rise,  or  a  slope  of 
33°42'  with  weights  per  cubic  foot  varying 
from  100  to  130. 

It  is  always  advisable,  where  practicable, 
to  put  a  layer  of  shingle  next  the  wall,  and 
to  consolidate  the  layers  of  the  filling  by 
punning  or  other  means,  so  as  to  reduce  the 
natural  slope  as  much  as  possible. 

With  a  well-built  wall,  designed  after 
methods  to  be  given  ;  having  a  good  foun- 
dation-course, larger  than  the  body  of  the 
wall,  to  better  distribute  the  pressure,  and 
resist  sliding,  and  backed  as  described  ;  with 
weeping  holes  near  the  bottom  at  intervals, 
—  there  should  be  no  fear  of  failure  under 
ordinary  conditions. 

6.  It  would  take  us  too  far  to  enter  into 
the  history  of  the  theory  of  the  retaining- 
wall.  On  this  point  see  an  interesting  article 
by  Professor  A.  J.  DuBois  in  the  "  Journal 


of  the  Franklin  Institute "  for  December, 
1879,  on  "  A  New  Theory  of  the  Retaining- 
Wall."  In  this  work^three  methods  will  be 
developed :  the  first,  a  graphical  method  that 
will  make  clear  the  foundations  on  which  all 
the  theory  rests;  the  second,  a  purely  ana- 
lytical method,  and  the  third,  a  graphical 
solution  founded  on  it.  Only  the  two  graph- 
ical methods  are  available  where  the  earth 
surface  is  not  plane. 

7.  In  case  a  wall  moves  forward,  how- 
ever little,  or  there  is  settling  of  the  earth 
behind  it,  the  earth  generally  rubs  against 
the  back  of  the  wall,  thus  developing  fric- 
tion. There  are,  however,  certain  inclina- 
tions of  the  back  of  the  wall  that  will  be 
specially  examined  in  articles  28-31,  for 
which  the  earth  sooner  breaks  along  some 
interior  plane,  in  its  mass,  than  along  the 
wall,  so  that  a  certain  wedge  of  earth  will 
move  with  the  wall  as  it  overturns  or 
tends  to  move.  For  all  other  cases,  which 
include  nearly  all  the  cases  in  practice, 
there  will  be  rubbing  of  the  earth  against 
the  wall,  so  that  the  earth-thrust  against 


8 


the  wall  must  be  assumed  to  make,  with 
the  normal  to  the  wall,  an  angle  equal  to  the 
co- efficient  of  friction  of  earth  on  wall, 
unless  this  is  greater  than  for  earth  on  earth, 
in  which  case  any  slight  motion  of  the  wall 
forward  will  carry  with  it  a  thin  layer  of 
earth,  so  that  the  rubbing  surfaces  are  those 
of  earth  on  earth. 

8.  These  suppositions  are  found  to  agree 
with  experiments.  The  old  theory  that 
assumed  the  earth- thrust  as  normal  to  the 
back  of  the  wall,  or,  as  in  Kankine's  theory, 
always  parallel  to  the  top  slope,  does  not  so 
agree,  and,  in  fact,  often  gives,  for  walls 
at  the  limit  of  stability,  the  computed  thrust 
as  double  that  actually  experienced.  The 
true  theory,  therefore,  includes  all  the  fric- 
tion at  the  back  of  the  wall  that  is  capable 
of  being  exerted.  This  friction,  combined 
with  the  normal  component  of  the  thrust, 
gives  the  resultant  earth-thrust  inclined 
below  the  normal  to  the  back  of  the  wall  at 
the  angle  of  friction  to  this  normal.1 


1  In  Annales  des  Fonts  et  Chaussees  for  April,  1887, 
M.  Siegler  has  given  the  results  of  some  simple  experiments 
proving  the  existence  of  a  vertical  component  of  the  earth- 


9.  Rankine's  assumption  that  the  direc- 
tion of  the  earth-thrust  is  always  parallel  to 
the  top  slope  applies  only  to  the  case  of  an 
imaginary  incompressible  earth,  homogene- 
ous, made  up  of  little  grains,  possessing 
the  resistance  to  sliding  over  each  other 
called  friction,  but  without  cohesion  ;  of  in- 
definite extent,  the  top  surface  being  plane  ; 
the  earth  resting  on  an  incompressible  foun- 
dation, or  one  uniformly  compressible,  and 


thrust  against  the  movable  side  of  a  box  filled  with  sand,  by 
actually  measuring  the  increased  friction  at  the  bottom  of 
the  movable  board,  held  in  place,  caused  by  this  vertical  com- 
ponent. The  box  was  one  foot  square  at  the  base ;  and  for 
successive  heights  of  sand  of  one-third,  two-thirds,  and  one 
foot,  the  vertical  components  of  the  thrust  for  earth  level  at 
top  were  0.66  pound,  1.76  pounds,  and  3.97  pounds,  respec- 
tively. Similarly  for  a  box,  0.5  x  0.8  feet,  filled  with  sand, 
but  having  a  movable  bottom  supported  firmly  on  iron  blocks, 
the  force  necessary  to  move  the  blocks  under  the  sides  and 
under  the  bottom  was  measured;  and  from  this  the  relative 
weights  of  sand  supported  by  the  bottom  and  sides  of  the 
box  was  found  to  be  as  one  to  one,  nearly,  for  a  height  of 
sand  of  0.6  foot,  and  about  two  to  one  for  a  height  of  1.18 
foot,  the  total  weights  ascertained  by  the  friction  apparatus 
also  checking  out  with  the  actual  to  within  five  per  cent. 
Other  experimenters  have  actually  weighed  the  amounts  held 
up  by  the  sides  and  bottom,  respectively.  See  Engineering 
News  for  May  15  and  29,  1886,  also  the  issue  for  March  3, 
1883,  on  "A  Study  of  the  Movement  of  Sand;"  also  see 
article  60  following. 


10 


being  subjected  to  no  external  force  but  its 
own  weight. 

For  such  a  material,  the  only  pressure 
which  any  portion  of  a  plane  parallel  to  the 
top  slope  of  greatest  declivity  can  have  to 
sustain  is  the  weight  of  material  directly 
above  it ;  so  that  the  pressure  on  the  plane 
is  everywhere  uniform  and  vertical.  If  we 
now  suppose  a  parallelopipedical  particle, 
whose  upper  and  lower  surfaces  are  planes 
parallel  to  the  top  slope,  and  bounded  on 
the  other  four  sides  by  vertical  planes,  we 
see  that  the  pressures  on  the  upper  and 
lower  surfaces  are  vertical,  and  their  differ- 
ence is  equal,  opposite  to,  and  balanced  by 
the  weight  of  the  particle.  It  follows  that 
the  pressures  on  the  opposite  vertical  faces 
of  the  particle  must  balance  each  other 
independently,  which  can  only  happen  when 
they  act  parallel  to  the  top  surface,  in  which 
case  only  are  they  directly  opposed.  The 
pressures,  therefore,  on  the  two  vertical 
faces  parallel  to  the  line  of  greatest  declivity 
will  be  horizontal ;  and  on  the  other  two 
faces,  parallel  to  the  line  of  greatest  de- 
clivity. This  is  Rankine's  reasoning,  and 


11 


it  is  sound  for  the  material  and  conditions 
assumed.  It  is  likewise  applicable  to  a 
material  of  the  same  kind,  only  compressible  * 
provided  we  suppose  it  deposited,  as  snow 
falls,  everywhere  to  the  same  depth,  on  an 
absolutely  incompressible,  or  a  uniformly 
compressible,  plane  foundation,  parallel  to 
the  ultimate  top  slope  of  the  earth  ;  for  then 
the  compression  is  uniform  throughout  the 
mass,  and  does  not  affect  the  reasoning. 
But  if  we  suppose,  as  usually  happens,  that 
the  foundation  is  not  uniform  in  compressi- 
bility, then  the  earth  will  tend  to  sink  where 
it  is  most  yielding.  This  sinking  is  resisted 
to  a  certain  extent  by  the  friction  resulting 
from  the  thrust  of  the  earth  surrounding  the 
falling  mass,  so  that  much  of  its  weight  is 
transmitted  to  the  sides,  as  actually  happens 
in  the  case  of  fresh  earth  deposited  over 
drains,  culverts,  or  tunnel  linings  which 
settle  appreciably.  In  the  case  of  a  tunnel 
driven  through  old  ground,  most  if  not  all 
the  weight  of  the  mass  above  it  is  trans- 
mitted to  the  sides  ;  at  least,  at  first,  before 
the  timbering  or  masonry  is  got  in.  Again, 
if  the  mass  of  earth  is  of  variable  depth, 


12 


even  on  a  firm  foundation,  the  mass  of 
greatest  depth  will  sink  most,  thus  trans- 
mitting some  of  its  weight  to  the  sides,  so 
that  throughout  the  entire  mass  the  press- 
ure is  nowhere  the  same  at  the  same  depth 
as  assumed.  The  vertical  pressure  over  a 
drain  or  small  culvert  crossing  an  ordinary 
road  embankment  is  less,  too,  for  another 
reason,  where  the  embankment  is  highest. 
The  earth-thrust  on  a  vertical  plane,  parallel 
to  the  line  of  road,  is  horizontal  for  a  sym- 
metrical section  when  the  plane  bisects  that 
section.  On  combining  this  thrust  with  the 
weight  of  the  material  on  either  side,  we 
see  that  the  resultant  load  on  the  culvert  is 
removed  farther  from  the  centre  than  if 
there  was  no  horizontal  thrust.  It  is  on 
account  of  this  tendency  to  equalize  press- 
ure by  aid  of  the  friction  resulting  from  the 
earth-thrust,  that  sand,  when  it  can  be  con- 
fined, is  one  of  the  best  foundations,  whether 
in  mass  or  in  the  form  of  sand  piles. 

10.  In  the  case  of  earth  deposited  behind 
a  retaining-wall  on  a  good  foundation,  the 
settling  of  the  earth  will  generally  be  greater 
than  that  of  the  wall,  so  that  the  earth  rubs 


13 


against  the  wall,  giving  generally  the  direc- 
tion of  the  thrust  no  longer  inclined,  even 
approximately  parallel  to  the  top  slope 
(except  when  the  latter  is  at  the  angle  of 
repose),  but  making  with  the  normal  to  the 
back  of  the  wall  an  angle  downwards  equal 
to  the  angle  of  friction.  If  the  wall  should 
settle  more  than  the  filling,  the  thrust  would 
at  first  have  a  tendency  to  be  raised  above 
the  normal.  But  if  such  a  thrust,  when 
combined  with  the  weight  of  the  wall,  passes 
outside  of  the  centre  of  the  base  of  the 
wall,  the  top  of  the  wall  will  move  over 
slightly,  the  earth  will  get  a  grip  on  the  wall 
in  the  other  direction  ;  so  that  it  is  plainly 
impossible  for  the  wall  (for  usual  batters  at 
least)  to  overturn  or  slide  on  its  base,  with- 
out this  full  friction,  acting  downwards  at 
the  back  of  the  wall,  being  exerted.  Hence 
the  theory  which  supposes  it  is  safe  ;  for 
although  it  is  possible  that  the  earth  may 
make  the  effort  at  times  to  exert  the  full 
thrust  given  by  Rankine's  formula,  yet  this 
effort  is  suppressed  instanter  by  the  external 
force  now  introduced  by  the  wall  friction, 
which  force  was  expressly  excluded  from 


14 


the    Rankine  theory.      The    exceptions    to 
this  rule  will  be  noted  in  article  31. 

11.  Weyranch's  objections  to  taking  the 
thrust  inclined   at  the  angle  <f>'  of   friction 
to  the  normal  are   easily  met.      He   says, 
Take  a  tunnel-arch ;  and  if  we  suppose  the 
pressure,  as  we  go  up  from  either  side,  to 
make  always  the  angle  <£'  with  the  normal, 
we  shall  have  at  the  crown  two  differently 
directed  pressures :  similarly  for  a  horizon- 
tal wall  with  level-topped  earth  resting  on 
it.     If  there  is  no  relative  motion,  or  ten- 
dency to   motion,   the  thrust    in   the  latter 
case  is  of  course  vertical,  and  in  the  former 
is  probably  vertical  at  the  crown   and  in- 
clined elsewhere  ;    but  if  the   arch  or  wall 
moves,  and  there  is  rubbing  of   the  earth 
on  the  masonry,  there  is  necessarily  friction 
exerted  ;  so  that  the  thrust  at  any  point  can 
have  but  one  direction,  making  the  angle  </>' 
with  the  normal. 

12.  Mr.  Benjamin   Baker,  in   his    paper 
before  the  Institution   of  Civil  Engineers, 
on  the  "  Actual  Lateral  Pressure  of  Earth- 
work "    (republished  by  Van    Nostrand  as 
"  Science  Series,"  No.  56),  tested  an  old 


15 


theory  (where  the  earth-thrust  was  assumed 
to  act  normal  to  the  wall)  by  the  results 
of  experiments,  and  found  the  theoretical 
pressure  often  double  the  actual.  In  the 
discussion  which  followed,  not  a  single 
engineer  so  much  as  alluded  to  a  truer 
theory  which  assumes  the  true  direction  of 
the  earth-thrust,  and  has  been  known  and 
used,  just  across  the  channel,  since  the 
time  of  Poncelet. 

The  writer  tested  this  theory  by  many  of 
the  experiments  recorded  by  Baker  and 
some  others,  and  found  it  to  agree,  within 
certain  limits,  remarkably  well  (see  "Van 
Nostrand's  Magazine  "  for  February,  1882). 
These  results  have  been  carefully  revised, 
and  new  experiments  included,  in  the  table 
given  farther  on,  from  which  the  reader 
can  form  a  fair  estimate  of  the  theory  as  a 
working  theory  within  certain  limits  that 
will  be  indicated. 

The  reader  is  referred,  however,  to  Mr. 
Baker's  essay,  not  only  for  experiences 
under  ordinary  conditions,  but  for  those 
exceptional  cases  which  seem  to  defy  all 
mathematical  analysis.  In  fact,  the  engi- 


16 


neer  almost  invariably  lias  to  assume  che 
weights  of  earth  and  masonry,  and  angie 
of  repose  of  the  earth.  Where  there  is 
water,  the  conditions  one  day  may  be  very 
different  from  what  they  are  the  next, 
especially  if  the  foundation  is  bad,  as  often 
happens  ;  in  which  case  the  wall  will  move 
over  simply  on  account  of  the  compres- 
sibility of  the  foundation,  so  that  it  has 
perhaps  nothing  like  the  estimated  stability. 
For  all  such  cases  an  allowance  must  be 
made  over  the  results  given  for  a  firm 
foundation,  etc.,  as  to  which  no  rule  can 
be  given. 

As  water  often  saturates  the  filling,  and 
perhaps  gets  under  the  wall,  we  must  con- 
sider, in  certain  cases,  water-pressure  in 
connection  with  the  thrust  of  the  backing. 
Therefore,  a  short  chapter  on  reservoir- 
walls,  or  dams,  follows,  in  which  many  of 
the  principles  that  must  likewise  apply  to 
retaining-walls  proper  are  given. 


CHAPTER  I. 

RESERVOIR-WALLS. 

13.  THE  design  of  reservoir- walls  is  a 
subject  that  has  received  the  attention  of 
many  engineers  and  mathematicians ;  but 
they  are  by  no  means  agreed,  except  in  a 
general  way,  upon  the  precise  profile  that 
is  best  to  satisfy,  as  uniformly  as  possible, 
the  requirements  of  strength  and  stability. 

We  shall  very  briefly,  and  by  the  shortest 
means,  point  out  the  main  principles  of 
design  of  a  dam  that  resists  overturning 
or  sliding  by  its  weight  alone,  and  is  called 
a  gravity  dam,  in  contradistinction  to  one 
built  on  a  curve  that  requires  the  aid  of 
arch  action  to  render  it  stable. 

Let  Fig.  1  represent  a  slice  of  the  dam 
contained  between  two  vertical  parallel 
planes  one  foot  apart,  and  perpendicular  to 
the  faces. 

When   the   dam   is   large,  a  roadway  is 


18 


generally  built  on  top,  so  that  the  faces  ks 
and  gi  are  vertical  or  nearly  so  for  some 
distance  down ;  after  which  the  profile  is 
designed  to  meet  certain  requirements,  to  be 
given  presently.  Let  us  suppose  that  the 


.«rr.TTTrrrrrrrrnft 


dam  has  been  properly  designed  down  to 
the  horizontal  joint  df/,  and  that  the  weight 
of  the  portion  above  df  equals  Wv  regard- 
ing the  weight  of  a  cubic  foot  of  masonry 
as  1,  and  that  its  resultant  cuts  the  joint 
df  at  the  point  o. 


19 


To  design  the  part  fabd  below  df  by  a 
rapid  though  tentative  method,  we  must 
first  assume  the  slopes  db  and  fa  corre- 
sponding to  the  depth  dc;  then  compute  the 
areas  of  the  triangles  bed  and  afe,  and  of 
the  rectangle  feed.  The  distances  of  the 
centres  of  gravity  of  these  areas  (which 
represent  volumes)  from  the  point  b  are  re- 
spectively §6c,  be  +  ^ae,  and  be  -j-  \ce.  On 
multiplying  each  area  by  its  correspond- 
ing arm  from  #,  adding  the  products 
to  W^bo  -f-  do),  and  dividing  by  the  sum 
of  Wl  (which  equals  the  area  of  gMf) 
and  the  portion  added  fabd,  we  find  the 
horizontal  distance  bm  from  b  to  where  the 
resultant  of  the  weight  above  joint  ab  cuts 
this  joint.  Its  amount  W  is  equal  to  the 
sum  of  the  areas  (  TF3  +  abdf),  and  we 
have  only  to  combine  W  acting  along  the 
vertical  through  w,  with  the  horizontal 
thrust  H  .rf  the  water  acting  on  the  face 
ksdb,  to  uud  the  resultant  R  on  the  joint, 
and  the  po»*jt  n  where  it  cuts  that  joint. 

There  is  a  vertical  pressure  of  the  water 
on  the  part  sdb;  but,  as  it  adds  to  the 
stability,  it  is  generally  neglected,  particu- 


20 

larly  as  the  inner  face  is  generally  nearly 
vertical. 

14.  The  horizontal  pressure  of  the  water 
H  for  the  height  7i,  by  -known  laws  of 
mechanics,  is  equal  to  the  area  h  x  1  mul- 
tiplied by  the  depth  of  its  centre  of  gravity 

-  below  the  surface  of   the  water,  and  by 

the  weight  of  a  cubic  foot  of  water  w> 
where  a  cubic  foot  of  masonry  is  taken  as 
the  unit.  This  pressure  acts  horizontally 
at  ^h  above  the  joint  afr,  so  that  its  moment 
about  the  point  n  where  the  resultant  R 

h          h        hsw     rp, 

cuts  the  base  db  is  h  .  -  -  w  •  ~  —  —TT-     l  M 
zoo 

moment  of  W  about  the  same  point  is 
W  X  mn.  As  these  two  moments  must  be 
equal,  we  find  the  distance  between  the 
resultant  pressures  on  joint  ab  for  reservoir 
empty  a'nd  reservoir  full, 

_  h*w 
=  GTf' 

The  above  is  substantially  one  of  the  methods 
adopted  by  Consulting  Engineer  A.  Fteley  in  the 
design  of  the  proposed  Quaker  Bridge  Dam.  See 
bis  interesting  report,  and  that  of  B.  S.  Church, 


21 


chief  engineer,  with  many  diagrams  of  existing 
dams  of  large  proportions,  in  "  Engineering  News  " 
for  1888,  Jan.  7,  14,  Feb.  4,  11;  also  the  discus- 
sions by  the  editor  in  the  numbers  for  Feb.  4  and 
25,  and  March  3. 

15.  There    are    three  well-known  condi- 
tions, that  must  hold  at  any  joint  if    the 
profiles   Ja    and    ~di>    have    been    designed 
correctly :  — 

1st,  The  points  m  and  n  where  the  re- 
sultants for  reservoir  empty  or  full  cut  the 
base  ab  must  lie  within  the  middle  third  of 
the  joint  or  base  ab. 

2d,  The  unit  pressures  of  the  masonry  at 
the  points  a  or  b  must  not  exceed  a  certain 
safe  limit. 

3d,  No  sliding  must  occur  at  any  point. 

16.  The    last    condition    is   evident,  and 
requires  that  H  <  Wf  where  /  is  the  co- 
efficient of  friction  of  masonry  on  masonry, 
the  adhesion  of  the  mortar  being  neglected. 
If  (f>  is  the  angle  of  repose  of  masonry  on 
masonry,  /  =  tan  <£,  and  we  must  always 
have, 


22 


that  is,  the  resultant  R  must  never  make 
with  the  normal  to  the  joint  an  angle 
greater  than  the  angle  of  friction.  In  fact, 
in  practice,  we  should  employ  some  factor 
of  safety  as  2  or  3,  so  that  2H  or  311  should 
always  be  less  than  Wf.  This  third  con- 
dition is  of  supreme  importance  at  the 
foundation  joints  of  dock-walls,  which  fail 
(wnen  they  fail  at  all)  by  sliding  from 
the  insufficient  friction  afforded  by  the  wet 
foundation.  For  ordinary  retaining- walls, 
too,  the  foundation  should,  when  practi- 
cable, be  inclined,  so  that  R  shall  make  a 
small  angle  with  the  normal  to  the  base. 
In  all  cases,  deep  foundations  are  to  be 
preferred,  as  the  earth  in  front  of  the  wall 
resists  the  tendency  to  slide  appreciably. 

17.  We  shall  now  proceed  to  give  a 
reason  for  the  first  condition  above,  and 
likewise  deduce  a  formula  to  ascertain  the 
unit  stresses  at  the  points  a  and  b. 

If  we  decompose  the  resultant  R  at  the 
point  w,  distant  u  =  an  from  a  (Fig.  1), 
into  its  two  components  H  and  TP,  the 
former  is  resisted  by  the  friction  of  the 
joint,  and  will  be  neglected  in  computing 


23 


the  stresses  at  a  and  fr,  though  it  doubtless 
affects  them  in  some  unknown  manner. 
The  remaining  force  W,  acting  vertically 
at  w,  must  necessarily  cause  greater  press- 
ure at  the  nearest  edge  than  elsewhere  on 
the  joint,  at  least  when  the  angle  at  a  is  not 
too  acute,  and  the  dam  is  a  monolithic 
structure.  For  large  dams  built  of  stones 
in  cement,  it  is  likely  that  there  will  be 
greater  pressure  at  the  middle  of  the  base 
than  in  a  monolithic  structure  where  the 
resistance  to  shearing  or  sliding  along  ver- 
tical planes  is  much  greater  than  in  a  wall 
made  up  of  many  blocks,  particularly  if 
they  are  laid  dry.  But  it  is  probably  best, 
until  experiment  can  speak  more  decisively 
on  the  point,  to  assume  the  pressure  great- 
est at  the  toe  nearest  the  resultant,  and  as 
given  by  the  following  theory  :  — 
Call  I  =  length  of  joint  ab 

u  =  an  =  distance  from  R  to  near- 
est toe  ; 

then  if  we  suppose  applied  at  the  centre  of 
the  joint  two  vertical  opposed  forces,  each 
equal  to  IF,  it  does  not  affect  equilibrium. 
We  can  now  suppose  the  force  W  acting 


24 


downwards  at  the  centre  to  be  the  resultant 

W 
of  a  uniformly  distributed  stress  p1  =  —  , 

shown  by  the  little  arrows  just  below  joint 
ab;  and  that  the  remaining  forces  TF,  one 
at  the  centre  and  one  at  n,  acting  in  oppo- 
site directions,  and  constituting  a  couple, 
whose  moment  is  W  (^l  —  u),  cause  a  uni- 
formly increasing  stress,  as  in  ordinary 
flexure  (shown  by  the  little  arrows  below 
the  first),  whose  intensity  at  a  or  b  is  by 
known  laws, 


The  total  stress  p  at  the  nearest  toe  a  is 
therefore  the  sum  of  pl  and  j?2,  and  is  com- 
pressive. 


The  stress  at  b  is  of  course  pl  —  p^  where 
this  is  not  minus  indicating  tension,  unless 
the  joint  can  stand  the  tension  required. 
If  we  call  u'  the  distance  from  n  to  the 
farthest  toe,  i.e.  u'  =  n&,  we  have  the  mo- 


25 


ment  of  the  two  weights  W=W(u'  —  $) . 
On  substituting  this  value  for  W(^l  —  u)  in 
the  value  for  p2  above,  we  find  for  the  unit 
stress  at  b  the  identical  equation  (1)  above, 
provided  we  replace  u  by  u' ;  so  that  the 
equation  is  general,  and  applies  to  either 
toe,  if  we  only  substitute  for  u  the  distance 
of  the  resultant  from  that  toe.  The  stress 
is  distributed,  as  shown  by  the  lower  set  of 
arrows  in  Fig.  1,  where  there  is  only  com- 
pression on  the  joint  as  should  always 
obtain.  The  stress  is  thus  uniformly  in- 
creasing from  the  right  to  the  left.  If  the 
limit  of  elasticity  is  nowhere  exceeded,  it 
follows  that  a  plane  joint  before  strain  will 
remain  a  plane  joint  after  strain,  as  must 
undoubtedly  be  the  rule  for  single  rectangu- 
lar blocks. 

Referring  to  equation  (1),  we  see  that  if 
we  replace  u  by  u'  =  f  Z,  that  the  stress  at  b 
is  zero,  from  which  point  it  increases  uni- 
formly to  a,  where  its  intensity,  for  u  =  JJ, 

W 

is  p  =  2—,  or  twice  the  mean.     For  greater 

i 

values  of  u'  than  f  Z,  the  stress  at  6  becomes 
tensile,  which  is  not  desirable ;  hence  the 


26 


reason  for  condition  1  above,  that  the  re- 
sultant should  lie  within  the  middle  third 
of  the  joint. 

If  the  joint  cannot  resist  tension  at  all, 
and  R  strikes  outside  the  middle  third,  the 
joint  will  bear  compression  only  over  a 
length  3«.,  and  the  maximum  intensity  at 

W 

a  is  now  2—.     This  is  evident,  if  we  treat 
3u 

3u  =  I'  as  the  length  of  joint,  and  substi- 
tute this  value  for  I  in  formula  (1).  There 
is  now  no  pressure  at  the  distance  3u  =  I' 
from  the  left  toe  by  the  previous  reasoning 
for  the  original  joint  /,  and  to  the  right  of 
that  point  the  joint  will  open,  or  tend  to 
open.  It  is  evident  for  full  security  that 
the  resultant  should  strike  within  the  mid- 
dle third  some  distance  to  allow  for  con- 
tingencies. 

18.  Having  computed  the  unit  pressures 
at  the  nearest  toes  for  reservoir  full  or 
empty,  condition  2  requires  that  these 
pressures  do  not  exceed  certain  limits  :  in 
case  they  do,  the  lower  profiles  have  to  be  re- 
vised, and  the  computation  above  repeated, 
until  all  the  conditions  are  satisfied. 


27 


In  the  proposed  design  for  Quaker  Bridge 
"Dam,  maximum  pressures  per  square  foot  at  the 
toes,  at  the  base,  were  limited  to  30,828  Ibs.  at 
the  back,  and  33,206  Ibs.  at  the  face;  these 
pressures  diminishing  gradually  to  one-half  to 
within  about  100  feet  from  the  top,  the  total 
height  of  dam  from  the  foundation  being  265 
feet ;  the  argument  being  that  the  lower  parts 
could  stand  more  pressure  than  the  upper  parts 
shortly  after  construction,  on  account  of  the 
cement  there  attaining  a  greater  strength.  Be- 
sides, for  this  unprecedented  height  of  dam,  to  keep 
the  lower  pressures  within  more  usual  limits  "it 
would  be  necessary  to  spread  the  lower  parts  in 
an  impracticable  manner,  and  to  incline  the  slopes 
to  an  extent  incompatible  with  strength." 

It  is  evident  that  by  this  method  of  design 
there  is  no  fixed  rule  by  which  any  two  computers 
could  arrive  at  the  same  profile,  having  given  the 
upper  part  empirically,  sufficient  in  section  to 
carry  a  roadway,  and  to  resist  the  additional 
stresses  due  to  the  shock  of  waves  and  ice,  at  a 
time,  too,  when  the  mortar  is  not  fully  set. 

Such  a  rule  is  most  easily  introduced  by 
requiring  a  certain  factor  of  safety  against  over- 
turning, and,  moreover,  that  the  factor  of  safety 
against  sliding  along  any  plane  shall  not  fall  below 
a  certain  amount.  It  is  suggested,  however,  that 
the  factors  of  safety  should  increase  from  the 
foundation  upwards,  to  make  the  section  equally 
strong  everywhere  against  overturning,  when 


28 


allowance  is  made  for  the  effects  of  wind  and  wave 
action,  floating  bodies,  the  expansive  force  of 
ice,  or  perhaps  the  malicious  use  of  dynamite. 
If  this  is  admitted,  it  would  add  one  more  con- 
•dition  (4)  to  the  three  previously  stated,  and 
•would  secure  greater  uniformity  in  design.  See 
Appendix. 

As  to  the  unit  pressure  test  (condition  2),  it 
must  be  observed,  that  we  know  little  or  nothing 
as  to  what  limit  to  impose;  for  not  only  is  the 
stress  all  dead  load  (which  would  allow  of  higher 
unit  stresses),  but  the  unit  resistance  of  masonry 
in  great  bulk  is  undoubtedly  much  greater  than 
in  small  masses  (not  to  speak  of  tests  on  small 
^specimens  as  a  criterion),  since  the  shearing  off 
'which  follows,  or  is  an  incident  to,  crushing  can 
hardly  occur  in  the  interior  of  a  large  mass  of 
masonry. 

19.  We  shall  find  in  the  end,  that,  for 
different  forms  of  retaining-walls  to  sustain 
earth,  that  a  factor  of  safety  of  about  2.5 
against  overturning  is  highly  desirable,  and 
that  it  will  generally  satisfy  the  middle  third 
limit.  In  such  walls  this  factor  must  be 
introduced  to  provide  against  an  actual 
increase  of  the  earth-thrust,  due  to  water, 
freezing,  accidental  loads,  and  above  all 
to  the  tremors  caused  by  passing  trains  or 


29 


vehicles  (if  these  are  not  considered  sepa- 
rately), which  it  is  well-known  have  caused, 
by  increased  weight,  and  the  increased 
pressure  due  to  lowering  the  natural  slope, 
a  gradual  leaning  and  destruction  of  walls 
of  considerable  stability  for  usual  loads. 

In  a  very  high  dam  this  is  different : 
the  pressure  rarely  changes  but  little,  ex- 
cept on  the  upper  portions  ;  so  that,  if  such 
conditions  were  to  hold  indefinitely,  the 
limiting  unit  stresses  should  control  the 
lower  profile  more  than  a  factor  of  safety 
against  overturning.  But,  as  pointed  out 
by  the  editor  of  "  Engineering  News  "  (in 
the  issues  above  referred  to),  a  dam  oq 
which  the  fate  of  a  city  may  ultimately 
depend  should  be  designed,  as  far  as  pos- 
sible, to  resist  earthquakes  also.  For  that 
contingency,  there  is  a  reason  for  the  factor 
of  safety  against  overturning  and  sliding 
being  as  great  as  possible  throughout ;  and 
by  putting  the  gravity  dam  in  the  arch 
form,  convex  up  stream,  the  resistance  to 
earthquake  and  other  shocks  is  enormously 
increased. 

20.  We  have  now  given  the  general  prin- 


30 


ciples  that  should  guide  in  the  design  of 
dams,  which  likewise  apply  in  the  design 
6f  retaining-walls  proper,  where,  however, 
the  height  is  rarely  sufficient  to  call  for 
much,  if  any,  change  of  profile,  and  the 
maximum  pressures  are  usually  far  within 
safe  limits  when  a  proper  factor  of  safety 
against  overturning  or  sliding  has  been 
introduced,  which  satisfies  likewise  the  con 
dition  that  the  resultant  shall  cut  the  base 
within  the  middle  third.  We  of  course 
have,  as  stated  before,  the  direction  of  the 
earth-thrust  inclined  below  the  normal  to 
the  wall  at  the  angle  of  friction  ;  otherwise, 
the  methods  above  are  applicable  when  the 
value  of  that  earth-thrust  has  been  deter- 
mined. For  dock  or  river  walls,  saturated 
with  water,  The  buoyant  and  lubricating 
effect  of  the  water  must  be  considered. 

If  we  suppose  the  filling  of  gravel,  the 
water  surrounding  each  stone  allows  free- 
dom of  motion  ;  but  the  weight  of  the  solid 
stones  of  the  filling  must  now  be  taken  less 
than  when  in  air,  by  the  weight  of  an  equal 
volume  of  water,  or  at  the  rate  of  62.4 
Ibs.  per  cubic  foot  (or  say  64  for  salt 


31 


water),  and  the  earth-thrust  then  found 
for  the  angle  of  repose  of  stone  lubricated 
with  water.  Thus,  if  the  weight  of  the 
solid  stone  be  150.4  Ibs.  per  cubic  foot, 
and  the  voids  are  thirty  per  cent,  the  weight 
of  solid  stone  in  water  is  88  Ibs.  per  cubic 
foot,  and  that  of  the  filling  88  x  .70  = 
61.6  Ibs.  in  water,  although  it  was  105  in 
air. 

If  the  wall  is  founded  on  a  porous 
stratum,  the  weight  of  the  masonry  is  sim- 
ilarly reduced  by  62.4  Ibs.  per  cubic  foot, 
or  say  one-half  ordinarily ;  but  if  the 
foundation  is  rock  or  good  clay,  "there  is 
no  more  reason  why  the  water  should  get 
under  the  wall  than  it  should  creep  through 
any  stratum  of  a  well-constructed  masonry 
or  puddle-dam,"  as  Mr.  Baker  has  ob- 
served. 

If  the  water  cannot  get  in  behind  the 
wall,  the  water  in  front  only  assists  the 
stability. 

It  has  been  previously  observed  that 
sliding  is  principally  to  be  guarded  against 
in  dock-walls  and  others  similarly  situated, 
which  can  only  be  done  by  a  sufficient 


32 


weight  of  masonry  irrespective  of  its  shape, 
unless  the  foundation  is  inclined,  which 
even  in  the  case  of  piling  has  been  effected 

Fig.  2 


by  driving  the  piles  obliquely,  of  course 
as  nearly  at  right  angles  to  the  resultant 
pressure  as  is  practicable. 

Fig.  2  represents  a  wall  with  a  curved 
batter,  in  brickwork  with  radiating  courses, 


33 


that  might  be  used  for  a  quay  or  river-wall, 
or  a  sea-wall,  as  ships  ean  come  closer  to 
the  brink  than  in  the  case  of  a  straight 
batter ;  besides,  for  sea-walls  it  resists  the 
action  of  the  waves  better.  The  centre  of 
gravity  can  be  found  by  dividing  the  cross 
section  up  into  approximate  rectilinear 
figures,  and  proceeding  as  in  finding  the 
position  of  W  in  Fig.  1.  Its  position  is 
a  little  farther  back  than  for  a  straight 
batter,  which  adds  to  its  stability.  But  it 
is  difficult  to  construct,  the  joints  at  the 
back  are  often  thicker  than  is  advisable, 
iuA-;.  there  is  probably  no  ultimate  economy 
ir  its  use. 


34 


CHAPTER   II. 

THEORY    OF    RETAINING- WALLS. 

Graphical  Method. 

21.  IN  the  theory  of  earth-pressure  that 
follows,  we  shall  consider  the  earth  as  a 
homogeneous,  compressible  mass,  made  up 
of  particles  possessing  the  resistance  to 
sliding  over  each  other  called  friction,  but 
without  cohesion.  This  is  a  much  simpler 
definition  than  the  one  that  Rankine's 
theory  calls  for  (see  Art.  9),  and  is  more 
true  to  nature;  the  only  approximation,  in 
fact,  consisting  in  neglecting  cohesion,  if 
we  consider  a  homogeneous  earth  like  drj 
sand. 

Let  Fig.  3  represent  a  vertical  section  of 
a  retaining-wall  ABCD,  backed  by  earth, 
whose  length  perpendicular  to  the  plane  of 
the  paper  is  unity. 


35 


Assumption.  We  assume  that  the  earth 
behind  the  wall,  whether  the  top  surface  is 
a  plane  or  not,  has  a  tendency  to  slide 
along  some  plane  surface  of  rupture  as 
Al,  A2,  .  .  .  . 

.  3 


No  proof  is  given  of  this  assumption,  so 
that  it  can  only  be  tested  by  experiment ; 
but  for  the  present  we  shall  adopt  it. 

In  connection  with  the  hypothesis  of  a, 
plane  surface  of  rupture,,  we  shall  use  only 
one  principle  of  mechanics  relative  to  the 


36 


stability  of  a  granular  mass,  first  stated 
by  Rankine  as  follows  :  — 

It  is  necessary  to  the  stabiliij  of  a 
granular  mass,  that  the  direction  of  the 
pressure  between  the  portions  into  lohich  it 
is  divided  by  any  plane  should  not,  at  any 
point,  make  ivith  the  normal  to  that  plane 
an  angle  exceeding  the  angle  of  repose. 

This  principle  will  alone  enable  us  to 
ascertain  the  earth-thrust  against  any  plane 
without  resorting  to  a  special  principle, 
like  Coulomb's  "  wedge  of  maximum 
thrust,"  which  last,  however,  will  be  in- 
cidentally demonstrated  as  a  consequence, 
of  the  above  law. 

22.  In  Fig.  3,  let  us  consider  the 
triangular  prisms  CMO,  CAl,  .  .  .  ,  as 
regards  sliding  down  their  bases  A0t 
Al,...  . 

If  AF  is  the  natural  slope  of  the  earth, 
the  tendency  of  the  prism  CAP  to  slide 
along  AF  is  exactly  balanced  by  friction, 
as  is  well  known.  But  if  we  consider 
other  possible  planes  of  rupture,  lying 
above  AF,  as  ylO,  Al,  .  .  .  ,  we  see,  unless 
the  wall  offers  a  resistance,  that  sliding 


37 


along  some  one  of  these  planes  must 
occur:  so  that, the  earth  exerts  an  active 
thrust  against  the  wall,  which  must  be 
resisted  by  it ;  otherwise,  overturning  or 
sliding  would  ensue. 

In  case  the  wall  is  subjected  to  a  thrust 
from  left  to  right,  as  /rom  earth,  water, 
etc.,  acting  on  BD,  and  this  thrust  is 
sufficient  to  more  than  counterbalance  the 
active  thrust  of  the  earth  to  the  right  of 
the  wall,  it  will  bring  in  the  passive 
resistance  of  the  earth  to  sliding  up  some 
plane  as  A'2,  and  the  surface  of  rupture 
will  now  resist  motion  upwards,  in  place  of 
downwards  as  hitherto. 

In  the  first  case,  of  active  thrust,  where 
the  prism  is  just  on  the  point  of  moving 
down  the  plane,  we  know  by  mechanics 
that  the  resultant  pressure  on  the  plane 
of  rupture  makes  an  angle  <£  cf  friction  of 
earth  on  earth  with  the  normal  to  that 
plane  and  directed  belotv  the  normal ;  in 
the  second  case,  of  passive  thrust,  the 
direction  of  the  pressure  lies  above  or 
nearer  the  horizontal  than  the  normal,  and 
makes  the  angle  <£  with  the  latter. 


38 


23.  In  the  first  case,  w.here  the  wall 
receives  only  the  active  thrust  of  the  prism 
of  maximum  thrust,  let  us  call  G  (Fig.  3) 
the  weight  in  pounds  of  this  prism  ;  S  the 
resultant  pressure  on  the  surface  of  rupture, 
making  an  angle  (f>  with  the  normal  to  that 
plane  below  the  normal ;  and  E  the  resultant 
earth-pressure  on  the  wall,  which  (except 
for  cases  to  be  noted  in  Art.  31)  makes  an 
angle  cf>'  of  friction  of  earth  on  wall  with 
the  normal  to  the  wall  below  the  normal, 
unless  (£'  >  (/>,  in  which  case  a  thin  layer 
of  earth  will  go  with  the  wall,  in  case  of 
relative  motion,  and  this  layer  rubbing 
against  the  remaining  earth  will  only  cause 
the  friction  of  earth  on  earth,  and  E  will 
only  be  directed  at  an  angle  $  below  the 
normal ;  supposing  always  that  the  tendency 
to  relative  motion  corresponds  to  the  earth 
moving  down  along  the  back  of  the  wall 
AC,  as  in  settling  from  its  compressibility, 
or  as  in  case  of  an  incipient  rotation  of  the 
wall  forward,  from  a  greater  pressure  on 
the  outer  toe  or  a  slight  unequal  compres- 
sion of  the  foundation. 

It  remains  to  find  the  position  of  the  true 


39 


plane  of  rupture.  As  preliminary  to  this, 
we  note  from  Fig.  3  an  expeditious  way  of 
finding  the  direction  of  S  on  any  trial  plane 
of  rupture,  as  A\.  Thus  calling  w  the  angle 
that  Al  makes  with  the  vertical  Al,  the 
straight  line  making  an  angle  (<£  -f-  w) 
with  any  horizontal,  as  DCI,  below  that 
horizontal,  is  parallel  to  S,  since  any  line 
inclined  at  an  angle  w  below  the  horizontal 
is  perpendicular  to  Al,  and  S  is  inclined  at 
an  angle  <£  below  that  normal.  In  laying 
off  the  equal  angles,  it  is  convenient  to  use 
a  common  radius,  AH,  to  describe  the  arcs 
having  A  and  I  respectively  as  centres,  and 
to  take  chord  distances  of  the  arcs  <£  and  to, 
and  lay  them  off  on  the  arc  with  /  as  a 
centre,  as  shown.  For  any  other  trial 
plane,  as  A2,  we  have  simply  to  lay  off  the 
corresponding  value  of  w  below  the  angle  (f> 
as  before. 

24.  We  shall  now  refer  to  Fig.  4,  to 
illustrate  the  general  method  to  follow  to 
find  the  earth-thrust  E  in  pounds.  Here 
the  wall,  one  foot  long  perpendicular  to  the 
plane  of  the  paper,  is  shown  in  section 
BACD,  the  earth  sloping  at  an  angle  from 


40 


some  point  on  the  top  of  the  wall  to  the 
point  marked  2,  where  it  is  horizontal. 
This  is  called  a  surcharged  \vall,  the  earth 


41 


lying  above  the  horizontal ,  plane  of  the  top 
of  the  wall  being  called  the  surcharge. 

Extend  the  line  AC  of  the  inner  face  to 
0,  where  it  intersects  the  top  slope  of  the 
earth  ;  the  possible  prisms  of  rupture  are 
then  .101,  .i02,  .403,  .  .  .  ,  and  we  shall 
now  proceed  to  reduce  these  areas  to  equiv- 
alent triangles  having  the  same  base  A2. 
Draw  the  parallels  00',  IT',  33',  .  .  .  ,  to 
line  A'2  to  intersection  with  a  perpendicular 
to  A"2,  passing  through  the  point  2.  Then 
the  triangle  .402  is  equivalent  to  the  triangle 
.40'2,  and  Al2  to  A\f'2,  so  that  triangle 
AO'l'  is  equivalent  to  .401.  Similarly  .426 
is  equivalent  to  triangle  A'2  6' A,  having  the 
same  base,  A'2,  and  vertices  in  a  line  parallel 
to  this  base,  giving  the  same  altitude.  Thus 
the  area  .4026.4  is  replaced  by  AQ'6'A  ;  and 
the  weight  of  the  corresponding  prism,  if  we 
call  e  the  weight  per  cubic  foot  of  earth, 
is  \A2  x  0'6'  X  e.  Similarly  the  weight  of 
.4024  is  ±A2  x  0'4'  X  e  ;  so  that  if  we  use 
O'l',  0'2,  0'3',  .  .  .  ,  to  represent  the  weights 
of  the  successive  prisms  .401,  ^402,  ^403> 
.  .  .  ,  on  the  force  diagram  given  below,, 
we  have  simply  to  multiply  the  value  of  J£, 


42 


given  by  construction,  by  %e,A'2  to  find  its 
true  value  in  pounds. 

We  next  lay  off  the  successive  values  of 
(<£  +  w),  as  in  Fig.  3.  Thus,  with  any 
convenient  radius,  as  .40,  we  describe  an 
arc,  ogdf,  and  call  the  intersections  with 
-41,  A'2,  .  .  .  ,  ar  a2,  a3,  .  .  .  ,  respectively. 
Next,  through  point  g  on  the  arc  in  the 
vertical  through  A,  draw  vertical  and  hori- 
zontal lines,  and  describe  an  arc,  bssv  .  .  . 
with  the  same  radius  ;  then  draw  gs,  making 
the  angle  <£  below  the  horizontal  gb  (by 
making  chord  bs  =  chord  fd) ,  and  lay  off 
with  dividers,  chords  ss^  ss2,  S6'3,  .  .  .  , 
•equal  to  chords  gar  ga^  ga^  ...  It  is 
•evident  now  that  lines  gsr  gs2,  gs3,  .  .  .  , 
make  the  angles  <£  with  the  normals  to  the 
successive  planes  ^41,^42,^.3,  .  .  .  ,  and  thus 
give  the  direction  of  the  >S's  corresponding 
to  those  planes. 

We  now  lay  off  with  dividers  on  the 
vertical  line  gA  the  distances  ggr  gg^  .  .  . 
equal  respectively  to  OT,  0'2,  0'3',  .  .  .  , 
and  draw  through  the  points  gv  g.2,  gs, 
parallels  to  the  direction  of  E  (drawn  as 
before  explained)  to  intersection  with  the 


43 


lines  f/Sj,,  </s.2,  gsa,  .  .  .  ,  which  intersections 
call  cr  c2,  c3,  .  .  .  ,  respectively. 

25.  It  follows  that  the  lines  g^  g<£.2, 
03c3,  .  .  .  ,  represent  the  thrusts  E  due  to 
the  successive  prisms  of  rupture  .401,  ./102, 
.  .  .  ,  and  we  shall  now  prove  that  the 
greatest  of  these  lines,  which  is  found  to 
be  0^,  represents  the  actual  active  thrust 
upon  any  stable  wall.1  This  follows  from 
the  simple  fact,  that  if  we  regard  any 
other  thrust  than  the  maximum  as  the  true 
one,  on  combining  this  lesser  thrust,  taken 
as  acting  to  the  right,  with  the  weight  of 
the  wedge  of  rupture  corresponding  to  the 
maximum  thrust,  we  necessarily  find  that 
the  resultant  falls  below  the  position  first 
assumed  ;  so  that  it  makes  an  angle  with  the 
normal  to  the  corresponding  plane  of  rupture 
greater  than  the  angle  of  repose,  which,  by 
the  principle  of  Art.  21,  is  inconsistent  with 
stability.  Thus,  in  Fig.  4,  if  we  choose 
tc  assert  that  any  trial  thrust,  as  ^2c2,  less 
than  the  maximum  04c4,  is  the  true  one,  on 

1  This  method  of  laying  off  the  trial  thrusts,  so  that  the 
maximum  could  readily  be  obtained,  was  first  given  by 
I'rofessor  Eddy,  in  New  Constructions  in  Graphical  Statics* 


44 


shortening  the  lengths  </3c3,  #4c4,  .  .  .  , 
representing  superior  thrusts,  to  the  com- 
mon length  <T/^,  and  drawing  through  the 
new  positions  of  c3,  c4,  .  .  .  ,  straight  lines 
to  g,  which  thus  represent  the  resultant 
thrusts  on  the  planes  ^43,  ^14,  .  .  .  ,  we  see 
that  the  new  directions  fall  below  the  first 
assumed  positions,  and  therefore  make 
angles  with  the  normals  to  the  planes 
greater  than  (£,  which  is  absolutely  incon- 
sistent with  equilibrium.  It  follows  that 
any  thrust  less  than  the  maximum,  as 
determined  by  the  construction  above,  is 
impossible  ;  and  that  this  maximum  thrust 
thus  found  is  the  actual  active  thrust  exerted 
against  the  wall.  In  this  consists  what  is 
known  as  Coulomb's  "  wedge  of  maximum 
thrust,"  which  is  here  established  by  aid  of 
the  single  mechanical  principle  enunciated 
in  Art.  21. 

The  prism  of  rupture  in  this  case  is 
^4024^4,  the  plane  AA  being  the  surface  of 
rupture. 

To  find  the  resultant  thrusts  on  all  the 
other  assumed  planes,  we  combine  the  actual 
thrust  found  with  the  weight  of  earth  lying 


45 


above  the  plane.  Thus,  extending  g^ 
.  .  .  to  a  common  length  #4c4,  or  to  the 
vertical  tangent  to  the  dotted  curve,  the  lines 
drawn  from  g  through  the  corresponding 
intersections  with  this  vertical  will  represent 
the  thrusts  on  the  planes  ^41,  ^42,  .  .  .  , 
which  are  thus  inclined  nearer  the  horizontal 
than  the  old  trial  values,  and  thus  make  less 
angles  than  c£  with  the  normals  to  their 
corresponding  planes  ;  so  that  the  conditions 
of  stability  are  all  satisfied,  and,  if  the  wall 
gives,  sliding  will  only  occur  down  the  plane 
of  rupture  A\. 

In  the  analytical  method  followed  by 
Wey  ranch,  E  is  assumed  to  be  constant, 
and  to  equal  the  actual  thrust  on  the  wall ; 
and  the  real  surface  of  rupture  is  taken  to 
be  that  plane  for  which  the  angle  that  S 
(Fig.  3)  makes  with  the  normal  is  the 
greatest  (<£)  consistent  with  equilibrium, 
which  is  in  agreement  with  what  we  have 
just  proved 

Winkler  adopts  the  same  method,  in 
preference  to  the  Coulomb  method.  In 
fact,  he  asserts  that  uno  author,  from 
Coulomb  down,  has  given  any  direct  satis- 


46 


factory  proof  of  Coulomb's  principle."  It 
is  hoped  that  the  above  demonstration 
will  prove  complete  and  satisfactory.  The 
method  evidently  gives  the  least  thrust,  for 
the  assumed  direction  of  E,  that  will  keep 
the  mass  from  sliding  down  the  surface  of 
rupture. 

The  earth  can  resist  a  much  greater 
pressure  from  the  wall  side,  since  a  con- 
tinuously increasing  pressure  from  the  left 
causes  all  the  resultants  on  planes  Al,  A'2, 
...  to  approach  the  normals,  then  to  pass 
them,  and  finally  to  lie  above  them,  with 
the  sole  condition  that  none  of  them  must 
make  angles  greater  than  $  with  their 
corresponding  normals  (see  Art.  34). 

26.  To  find  the  thrust  E  in  pounds,  we 
multiply  g4c4  to  scale  by  ^A2.e.  Finally, 
if  we  know  the  position  of  E,  we  combine 
it  with  the  weight  of  the  wall  in  pounds, 
acting  along  the  vertical  through  its  centre 
of  gravity,  to  get  the  resultant  on  the  base. 
If  the  upper  surface  of  the  earth  is  level, 
or  with  a  uniform  slope  from  the  point  0 
(Fig.  4),  then  the  sections  of  the  prisms  of 
rupture  for  various  heights  of  the  wall,  or 


47 


for  any  values  of  AQ,  are  similar  triangles, 
so  that  the  thrusts  E,  which  vary  directly 
with  the  weight  of  the  corresponding  prisms, 
will  also  vary  directly  as  the  areas  of  these 
triangles,  or  as  the  squares  of  the  homologous 
lines  .40,  or  as  the  squares  of  the  height  of 
point  0  from  the  base  AB.  It  follows,  as 
in  the  case  of  water-pressure,  that  for  these 
cases  the  resultant  E  of  the  earth-thrust 
acting  along  the  face  AQ  is  found  at  a 
point  | AO  along  AO  from  the  base  AB. 

For  the  surcharged  wall  it  is  possibly 
higher ;  in  fact,  Scheffler  takes  it  in  con- 
structing his  tables,  for  all  cases,  at  T% AQ 
along  AO.  But  experiment  indicates  either 
that  the  thrust  is  overestimated  for  sur- 
charged walls,  or  that  it  acts  not  higher 
than  at  one-third  the  height  of  0  above  the 
base  ;  so  that  it  will  prove  safe  in  practice 
to  take  the  latter  limit  if  we  use  the 
theoretical  thrust.  As  to  the  latter,  it  is 
evident  that  cohesion  (which  we  have 
neglected)  has  a  greater  area  to  act  upon 
along  the  surface  of  rupture  for  any  kind 
of  surcharged  wall,  than  for  earth  either 
level  or  sloping  down  from  the  top  of  the 


48 


wall ;  so  that  we  should  expect  the  thrust  to 
be  somewhat  overestimated  when  we  neglect 
cohesion  altogether,  since  the  resistance  to 
sliding  down  any  plane  due  to  it  is  directly 
as  the  area  of  the  surface  of  separation. 

27.  In  case  the  earth  is  level  with  the 
top  of  the  wall,  the  construction  of  Fig.  4 
again  applies,  only  the  line  0'6'  now  coincides 
with  the  horizontal  through  (7,  and  the 
reduction  of  areas  to  equivalent  triangles 
is  omitted,  since  now  all  the  triangles  have 
the  same  altitude,  equal  to  the  height  of  the 
wall. 

If,  however,  the  earth  slopes  uniformly 
from  the  top  of  the  wall,  at  a  less  angle 
than  the  angle  of  repose,  we  can  assume 
any  point  as  2,  on  this  slope,  and  effect  the 
construction  of  Fig.  4  as  before ;  or,  better, 
we  can  divide  this  slope  into  a  number  of 
parts  at  1,  2,  .  .  .  ,  and  treat  01,  02,  .  .  .  , 
successively  as  the  bases  and  the  perpen- 
dicular from  A  upon  02,  produced  as  the 
common  altitude;  so  that,  using  01,  02, 
.  .  .  ,  as  representing  the  weights  of  the 
corresponding  prisms  on  the  load  line  gg, 
we  have  finally  to  multiply  the  value  of  </c, 


49 


corresponding  to  the  greatest  thrust,  by 
\e,  multiplied  by  this  perpendicular,  to  get 
the  maximum  thrust  E  in  pounds. 

In  case  the  surface  of  the  earth  slopes 
indefinitely  at  the  angle  of  repose,  the 
graphical  method  fails  to  find  the  surface 
of  rupture,  which  analysis  shows,  in  this 
case,  to  approach  indefinitely  to  the  plane 
of  natural  slope  passing  through  the  point 
A,  though  practically  it  may  be  shown  that 
planes  of  rupture  slightly  above  the  latter 
will  give  almost  identically  the  same  earth- 
thrust,  so  that  they  can  safely  be  used.  In 
fact,  it  is  well  to  state  here,  that,  for  earth- 
level  at  top,  the  surface  of  rupture,  as 
observed  in  experiments  with  every  kind  of 
backing,  agrees  very  well  with  theory ;  but, 
as  the  surcharge  grows  higher,  the  actual 
surface  of  rupture  lies  nearer  the  vertical 
than  the  theoretical,  and  the  thrust  is 
correspondingly  less,  particularly  for  walls 
leaning  backwards  at  top,  which,  for  a  high 
surcharge,  actually  receive  much  less  thrust 
than  the  nmple  theory  after  Coulomb's 
hypothesis,  neglecting  cohesion,  calls  for; 
ind  it  is  not  surprising  that  it  is  so.  But 


50 


we  shall  defer  the  comparison  of  numerical 
results  till  later. 

28.  Case  where  E  does  not  make  the  angle 
<£  or  <£'  with  the  normal  to  the  wall. 

In  Fig.  5,  let  AC  represent  the  inner  face 
of  the  wall,  backed  by  earth  sloping  upwards 
from.  C  in  the  direction  C  —  10.  There 
are  certain  positions  for  the  wall  AC  lying 
to  the  left  of  the  vertical  Ag,for  ivhich  the 
true  thrust  on  it  is  found  by  ascertaining 
the  thrust  on  the  vertical  plane  AO,  extending 
Jrom  the  foot  of  the  tuall  A  to  where  it 
intersects  the  top  slope  C  —  10,  having 
assumed  the  direction  of  the  thrust  on  AO, 
after  Rankine,  as  parallel  to  the  top  slope, 
and  combining  this  thrust,  acting  at  ^AO 
above  A,  with  the  weight  of  the  mass  of 
earth,  AOC,  lying  betiveen  AO  and  AC, 
acting  along  the  vertical  through  its  centre 
of  gravity.  The  thrust  on  AO  is  thus 
combined  with  the  weight  of  AOC,  at  a 
point  on  AC,  one-third  of  its  length  going 
from  A  to  C. 

This  direction  of  the  thrust  on  AO  par- 
allel to  the  top  slope  is  in  agreement  with 
Rankine's  principle  for  the  case  of  an 


51 


unlimited  mass  of  earth  of  the  same  depth 
everywhere,  on  an  uniformly  compressible 
foundation  (Art.  9),  and  doubtless  agrees 
very  nearly  with  the  direction  and  amount 
of  the  earth-thrust  in  ordinary  cases,  except 
near  comparatively  rigid  retaining-walls,  or 
other  bodies,  where  the  direction  is  generally 
changed,  as  previously  pointed  out. "  Let 
us  ascertain  the  limiting  position  of  AC, 
below  which  the  true  thrust  must  be  ascer- 
tained in  the  manner  just  stated.  To  do 
this,  we  first  assume  the  thrust  on  AO  as 
acting  parallel  to  the  top  slope,  and  find 
its  intensity  corresponding  by  previous 
methods  ;  and  afterwards  prove,  for  positions 
of  AC  below  the  limit,  to  be  found  by 
construction,  that  no  thrust  on  AO  having 
a  less  inclination  to  the  vertical  is  consistent 
with  equilibrium. 

The  construction  necessary  to  find  the 
thrust  on  AO,  from  the  earth  on  the  right, 
is  similar  to  that  given  for  Fig.  4,  except 
that  the  top  slope  is  now  uniform,  and  will 
only  be  briefly  indicated.  Thus,  divide  the 
top  slope  0  —  10,  to  the  right  of  Ag<  into  a 
number  of  parts,  made  equal  for  convenience, 


52 


and  draw  through  the  points  of  division  lines 
from  A  produced  on  to  meet  the  arc  described 
wiihAg  as  a  radius  at  the  points  ar  «2,  .  .  . 
Then  with  g  as  a  centre,  and  g A  as  a  radius, 
describe  a  semicircle  as  shown  ;  also  draw 
cjb  horizontal,  and  lay  off  arc  bs  equal  to  <£, 
the  angle  of  repose,  and  from  s  lay  off  arcs 
ssv  ss.2,  .  .  .  ,  equal  to  gav  ga2,  and  draw 
the  lines  gsv  gs^  .  .  .  ,  from  g  through 
the  extremities  of  these  arcs  to  represent  the 
directions  of  the  resultants  on  the  successive 
planes  of  rupture,  which  are  thus  inclined 
below  the  normals  to  those  planes  at  the 
angle  cf>  respectively.  Next,  on  the  vertical 
#3.,  lay  off  ggr  gg.2,  .  .  .  ,  equal  to  the  bases 
01,  02,  of  the  supposed  prisms  of  rupture 
lying  to  the  right  of  Ag,  and  through  their 
extremities  draw  g^V,  g.22',  .  .  .  ,  parallel 
to  top  slope  to  intersection  1',  2',  .  .  .  , 
with  the  directions  of  the  resultants  first 
found.  The  greatest  of  these  lines  ccx,  to 
scale,  represents  the  actual  thrust  OD  ^40  ; 
and  we  have  only  to  multiply  it  by  \ep, 
where  p  is  the  perpendicular  let  fall  from 
A  on  the  top  slope  0—10  produced,  to  scale, 
to  get  the  pressure  in  pounds,  if  desired. 


53 


Now,  if  the  direction  of  the  pressure  on  the 
Wall  AC  cannot  be  taken  as  usual,  inclined 
below  the  normal  to  AC,  at  an  angle  <£, 
it  is  (Art.  7)  because,  in  case  of  motion, 
the  earth  does  not  rub  against  the  wall 
sufficiently  to  develop  the  required  friction, 
whence  it  must  follow  that  the  earth  breaks 
along  some  plane  as  .44,  Ao,  .  .  .  ,  to  the 
left  of  Ag,  where  the  thrust  is  inclined  at 
the  angle  <£  to  its  normal ;  so  that  this  plane 
is  a  veritable  plane  of  rupture,  and  its 
position  can  be  found  as  usual  on  assuming 
the  direction  of  the  thrust  on  AQ  as  parallel 
to  the  top  slope. 

In  case  such  a  plane  exists  between  .40 
and  AC,  the  earth  below  it,  if  the  wall 
moves,  will  go  with  the  wall ;  further,  it  is 
evident  that  the  thrust  against  the  vertical 
plane  AO,  due  to  the  wedge  of  rupture  on 
the  left,  must  exactly  equal  the  thrust  first 
found  corresponding  to  the  wedge  of  rupture 
on  the  right,  otherwise  equilibrium  will  be 
impossible. 

To  ascertain  the  position  of  this  plane  of 
rupture  on  the  left,  that  we  shall  hereafter 
call  the  limiting  plane,  most  accurately,  it 


54 


is  weK  to  magnify  the  lines  representing  the 
forces  as  much  as  the  limits  of  the  drawing 
will  admit  of.  We  have  consequently 
divided  the  top  surface,  06',  into  a  number  of 
equal  parts,  of  which  the  first  eight  are  only 
one-fourth  the  length  of  the  corresponding 
parts  to  the  right  of  ^40.  By  laying  off  the 
loads  gg^  gg2,  .  .  .  ,  however,  to  a  scale 
four  times  as  large  as  just  used,  we  have 
the  lengths  gg^  gg.2,  .  .  .  ,  exactly  four 
times  the  lengths  01,  02,  .  .  .  ,  along  the 
surface  to  the  left  of  0,  so  that  the  old 
lettering  applies  again. 

We  now  produce  the  lines  A\,  .42,  .  .  .  , 
to  intersection  nv  n2,  .  .  .  ,  with  the  arc  gn 
(it  is  obvious  that  the  top  slope,  0(7,  should 
best  be  drawn,  in  the  first  instance,  through 
g,  for  accurately  fixing  the  positions  of  nv  n2, 
.  .  .)  ;  then  lay  off,  below  the  horizontal, 
the  angle  clgm  =  <£  ;  and  from  m,  the  inter- 
section of  gm  with  the  semicircle  cL46,  lay 
off  the  arcs  mmv  mw.2,  .  .  .  ,  equal  to 
gnv  gn2,  .  .  .  :  so  that  the  lines  gmr  gm2, 
o  .  .  ,  all  make  angles  equal  to  <£  with  the 
normals  to  the  corresponding  planes  Al, 
A2,... 


55 


Next,  on  drawing  through  gv  gv  .  .  .  , 
lines  parallel  to  the  assumed  direction  of 
the  thrust  on  AO,  to  intersection  with  the 
corresponding  lines  gmv  gm^  .  .  .  ,  the 
greatest  of  the  intercepts  (<75£5  nearly) 
represents,  to  the  scale  of  loads,  the  thrust 
on  the  plane  ^40  ;  and  this  length  should 
exactly  equal  four  times  the  length  cc7 
representing  the  thrust  from  the  right,  as 
we  find  to  be  the  case.  The  plane  of 
rupture  to  the  left  of  the  vertical  through 
A  thus  coincides  nearly  with  ^45,  which  is 
marked  "limit"  on  the  drawing.  [On  a 
larger  drawing,  for  c/>  =  33°  42'  and  the  top 
sloping  at  25°,  the  limiting  plane  was  found 
to  make  an  angle  of  15°  to  16°  (see  a  more 
accurate  determination  in  Art.  41)  to  the 
left  of  the  vertical  Ag,  and  to  lie  slightly 
below  ,45,  as  this  drawing  would  indicate.] 

If  we  lay  off  along  the  lines  parallel  to 
top  slope,  through  gv  g^  .  .  .  ,  the  true 
thrusts,  gjv  g.2t2,  .  .  .  ,  gjv  .  .  .  ,  the 
directions  of  gtr  gt^  .  .  .  ,  gtn,  .  .  .  ,  of 
the  true  thrusts  on  the  planes  A\,  ^12, 
.  .  .  ,  ^47,  .  .  .  ,  all  necessarily  lie  above 
the  first  assumed  directions ;  so  thaf,  the 


5G 


actual  thrusts  on  all  planes  other  than  Ao 
(which  we  shall  regard  as  the  plane  of 
rupture,  for  convenience),  lying  above  or 
below  ylo,  make  less  angles  with  the 
normals  to  those  planes  than  the  angle  of 
friction,  just  as  we  found  in  Art.  25. 

The  conditions  of  stability  of  Art.  21  are 
thus  satisfied  in  the  present  case  ;  but  it  is 
evident  that  this  is  no  longer  so  if  we  lower 
the  direction  of  the  thrust  on  ^fo,  which 
lessens  the  horizontal  component  of  the 
thrust  from  the  right,  since  intersections 
like  6',  7',  in  the  right  diagram  move  towards 
the  vertical  Ag,  though  the  reverse  obtains 
for  the  diagram  to  the  left,  which  of  itself 
indicates  some  absurdity.  If,  now  we 
combine  the  new  thrust  on  AQ  from  ?-he 
right  (which  has  a  less  horizontal  compo- 
nent than  before)  with  the  wedges  of  earth 
lying  to  the  left  of  AS,  it  is  readily  seen 
that  the  directions  of  some  of  the  resultants, 
as  gt^  .  .  .  ,  will  fall  below  their  first 
positions,  and  will  thus  make  greater  angles 
with  the  normals  to  their  planes  than  the 
laws  of  stability  will  admit  of ;  so  that  any 
lowering  of  the  first  assumed  position, 


57 


parallel  to  the  top  slope,  of  the  thrust  on 
./40,  is  impossible. 

We  thus  reduce  to  an  absurdity  every 
other  case  but  the  one  assumed,  which  is 
therefore  true ;  so  that  the  proposition 
enunciated  at  the  beginning  of  this  article 
is  demonstrated. 

We  see,  therefore,  that  we  cannot,  as 
before,  assume  the  direction  of  the  thrust 
on  the  wall,  AU,  as  having  the  direction 
gmc,  making  the  angle  (/>  with  the  normal 
to  AJC)  and  find  the  wedge  of  maximum 
thrust  corresponding ;  but  that  its  true 
direction,  gtc,  is  found  by  combining  the 
thrust  found  on  AO,  acting  parallel  to  top 
slope,  with  the  weight  of  the  wedge  of 
earth,  QAC,  between  the  wall  and  the 
vertical  plane  AO ;  otherwise,  if  the  left 
diagram  is  constructed,  we  find  its  direction 
and  amount  in  a  similar  manner  to  that 
used  in  finding  the  direction,  etc.,  of  gL, 
.  .  .  ,  by  laying  off  on  gA  (produced  if 
necessary)  (K7  X  4  ;  from  the  end  of  this 
line  we  draw  a  parallel  to  the  top  slope  (K) 
to  intersection  tc,  with  the  vertical  through 
t&.  The  line  gtc  to  the  last  scale  used  mui- 


58 


tiplied  by  %ep  (where  p  is  the  perpendiculai 
from  A  on  0(7  to  the  scale  used  in  laying  off 
06y)  gives  the  thrust  E  against  the  wall  in 
pounds.  It  is  laid  off  in  position  by  drawing 
a  line  parallel  to  gte  through  a  point  on  AU, 
%AC  above  -4,  as  previously  enunciated. 

29  (<f>' <</>).  In  case  this  construction 
gives  a  thrust  on  the  wall  which  makes  a 
greater  angle  with  its  normal  than  the 
co-efficient  of  friction,  <£'  of  wall  on  earth, 
(f)f  being  less  than  <£,  then  it  is  correct  to 
assume  the  direction  of  E  as  making  this 
angle  <£'  with  the  normal,  and  proceed  as  in 
Fig.  4  to  find  the  thrust.  In  the  preceding 
'article,  no  trial-thrust  on  the  vertical  plane 
A\J  was  assumed  to  lie  nearer  the  horizontal 
than  the  top  slope,  as  there  was  no  reason 
for  considering  such  exceptions  to  the  usual 
direction  in  a  mass  of  unlimited  extent. 
Now,  however,  the  wall  requires  the  thrust 
on  AO  to  lie  nearer  the  horizontal  than  0(? 
does,  in  which  case  the  horizontal  component 
will  be  increased  (since  intersections  like 
7',  8',  move  away  from  the  vertical  J#), 
and  the  thrusts  on  all  planes  Al,  A2,  .  .  .  , 
lying  to  the  left  of  Ag,  will  be  raised  above 


59 


their  previous  positions,  gtl9  gt^  .  .  .  ;  so 
that  the  thrusts  on  all  the  planes  now 
make  less  angles  than  0  with  the  normals 
to  those  planes,  so  that  the  conditions 
for  stability  of  "the  granular  mass "  are 
assured. 

30.  The  "limiting  plane,"  corresponding 
to  the  plane  of  rupture  on  the  left,  can  be 
found  by  a  different  construction  from  that 
given  above.  Thus,  having  found  the  line 
cc7  representing  the  maximum  thrust  from 
the  earth  to  the  right  of  -40,  multiply  by  4, 
say,  and  combine  with  the  successive  wedges 
of  earth  lying  to  the  left  of  ^40,  on  magni- 
fying the  lines  OT,  02,  .  .  .  ,  in  the  same 
proportion,  thus  giving  the  lines  gtv  gt^ 
.  .  .  ,  for  the  direction  of  the  thrusts  on 
the  planes  A1,A2,  .  .  .  ;  these  all  lie  above 
the  directions  gmr  #m2,  .  .  .  ,  making  the. 
angles  (f>  with  the  normals  to  the  planes, 
except  for  the  limiting  plane,  where  gt&  and 
gmb  nearly  coincide,  as  they  should  exactly 
if  A5  was  the  limiting  plane.  The  lowest 
relative  position  of  gt  with  respect  to  gm  is, 
of  course,  the  one  selected.  It  is  evident, 
though,  that  the  construction  for  the  wedge 


60 


of  greatest  thrust  to  the  left  of  Jig  gives 
a  more  accurate  evaluation  of  the  thrust 
than  the  one  to  the  right ;  so  that  we  can 
preferably  use  the  left  construction  not 
only  for  getting  the  limiting  plane,  but  for 
finding  the  thrust  on  any  wall  lying  below 
the  limiting  plane. 

It  is  evident,  from  what  precedes,  that 
the  double  construction  of  Art.  28  applies 
only  when  the  thrust  on  AQ  is  parallel  to 
the  top  slope  ;  for  the  moment  it  is  lowered, 
there  results  several  planes  of  rupture  to 
the  left  of  ^40,  which  is  impossible.  Even 
if  we  attempt  the  left  construction,  we  have 
seen  besides  that  the  resulting  thrust  on  ^40 
is  greater  than  by  the  construction  on  the 
right. 

In  case  the  face  of  the  wall,  AC,  lies 
above  the  "limiting  plane,"  as  found 
before,  we  evaluate  the  thrust  on  it,  as  in 
Fig.  4,  by  assuming  its  direction  to  make 
an  angle  with  the  normal  equal  to  </>  or  to 
<f>  when  </>'  <  (f).  Thus,  if  the  inner  face  of 
the  wall  had  the  position  y!2,  to  the  left  of 
-40,  the  direction  of  the  thrust  on  it  would 
now  be  gm^  in  place  of  gt^  as  before,  and 


61 


the  conditions  of  stability  of  the  granular 
mass  will  be  found  to  be  everywhere  verified 
as  in  Fig.  4  (see  Art.  25). 

31.  Summary.  —  For  all  cases  of  top 
slope,  when  the  inner  face  of  the  wall  is 
battered,  we  first  find  the  limiting  plane  by 
the  construction  of  Art.  28  ;  then  when  the 
inner  face  of  the  wall  makes  a  less  angle 
with  the  vertical  than  the  limiting  plane  does 
(as  is  nearly  always  the  case  in  practice, 
unless  the  surface  of  the  earth  slopes  at  or 
near  the  angle  of  repose,  in  which  case  the 
limiting  plane  is  at  or  very  near  the  vertical), 
we  assume  the  direction  of  the  thrust  on  it 
as  making  the  angle  <£  or  <£'  (for  <£'  <  <£)  with 
the  normal,  and  proceed  as  in  Art.  23,  et 
seq. ;  but,  if  the  face  of  the  wall  lies  below 
the  limiting  plane,  we  proceed  as  in  Art.  28, 
or  if  <f>f  «f>  we  .may  have  to  proceed  as  in 
Art.  29,  to  find  the  true  thrust. 

If  the  wall  leans  backward,  there  is  no 
need  to  find  the  limiting  plane,  as  the  usual 
construction  applies. 

For  earth  level  at  top,  the  limiting  plane 
is  inclined  to  the  left  of  the  vertical  equally 
with  the  plane  of  rupture  to  the  right ;  as 


62 


the  top  slope  increases,  it  approaches  the 
vertical,  and  coincides  with  it  for  the  surface 
sloping  at  the  angle  of  repose. 

Remark.  —  It  is  found  from  the  con- 
struction to  the  right  of  Ag,  in  Fig.  5,  for 
planes  of  rupture  lying  7°  to  14°  above 
the  one  corresponding  to  the  greatest  thrust, 
that  the  thrust  is  less  only  by  from  6  to  16 
per  cent,  though  it  more  rapidly  diminishes 
as  the  assumed  plane  of  rupture  nears  the 
vertical.  It  must  not  be  inferred,  then, 
particularly  for  steep  surface  slopes,  that 
a  considerable  divergence  between  the 
theoretical  and  actual  surfaces  of  rupture 
will  invalidate  the  theory,  if  the  object  is 
simply  to  get  the  thrust  within  a  few  per 
cent  of  the  truth,  particularly  as  the  theory 
neglects  cohesion.  In  fact,  for  a  surface 
slope  equal  to  the  angle  of  repose,  the  plane 
of  rupture  is  parallel  to  the  surface  ;  but  a 
plane  lying  much  nearer  the  vertical  will 
give  nearly  the  same  thrust. 

32,  In  this  connection,  it  may  be  well  to  describe 
an  experiment  made  by  Lieut.-Col.  Aude  in  1848, 
and  repeated  subsequently  by  Gen.  Ardant,  M. 
Curie,  and  M.  Gobin,  on  a  peculiar  retaining-wall 


63 


made  of  a  triangular  block  or  frame,  in  which 
the  inner  face  was  inclined  to  the  horizontal  at  the 
angle  of  repose  of  the  sand  backing,  when,  of 
course,  by  the  usual  assumption  as  to  E  making 
an  angle  of  <p  with  the  normal  to  the  wall,  the 
direction  of  E  would  be  vertical,  and  there  should 
be  no  horizontal  thrust!  This  seemed,  to  the 
French  experimenters,  to  offer  a  puzzling  objection 
to  theory;  but  the  solution  is  clearly  as  indicated 
in  Art.  28  (see  Art.  67,  Exps.  9  and  10).  Scheffler 
indicated  the  correct  solution  as  far  back  as  1857, 
but  gave  the  wrong  reason  for  it;  viz.,  that  the 
horizontal  thrust  was  thereby  greater. 

The  writer,  in  "  Van  Nostrand's  Magazine  "  for 
February,  1882  (p.  99),  pointed  out  that  any  other 
solution  than  that  indicated  in  Art.  28  was 
inconsistent  with  the  stability  of  a  granular  mass, 
and  the  computations  upon  that  basis  agreed  very 
closely  with  the  experiments.  Later  M.  Boussinesq 
has  developed  the  theory  of  the  limiting  plane  in 
connection  with  the  attempt  to  complete  the 
Rankine  theory,  by  considering  the  influence  of 
the  wall  on  the  pressures  even  to  a  finite  distance 
from  it.  According  to  Flamant,  he  defines  two 
limits  to  the  thrust,  and  considers  the  most  probable 
value  the  smaller  of  these  limits  augmented  by 
&  of  their  difference.  From  an  examination 
of  the  numerical  values  computed  by  Flamant 
("Annales  des  Fonts  et  Chausses,"  April,  1885), 
the  results  do  not  differ  greatly  from  those  given 
by  the  simple  theory  alone  used  in  this  work. 


64 


33.  The  disturbing  influence  of  the  wall  it- 
changing  the  normal  character  of  the  stresses  can 
be  illustrated  as  follows:  If  the  thrust  on  the 
vertical  plane  Aft  (Fig.  5)  acts  parallel  to  the 
surface  0  —  10,  it  meets  the  plane  of  rupture  at 
one-third  of  its  length  above  A,  through  which 
point  the  weight  of  the  prism  of  rupture  acts 
also;  so  that  the  resultant  on  this  plane  acts  at 
this  point,  which  corresponds  to  a  pressure  on 
the  plane  of  rupture  uniformly  increasing  from  the 
surface  downwards.  If,  however,  the  wall  causes 

the  thrust  on  AQ  to  make  as  [  angle  with 

i  greater  ) 

the  horizontal,  the  resultant  on  the  corresponding 

plane  of  rupture  on  the  right  acts    <      '  °W  [    the 

(  above  ) 

point  situated  at  one-third  of  the  length  of  the 
plane  above  A,  so  that  the  pressures  on  it  are  no 
longer  uniformly  increasing.  This  abnormal  state, 
doubtless,  does  not  extend  far  into  the  mass  before 
the  usual  direction  of  the  thrust  in  a  large  mass  of 
earth  is  attained ;  but  the  fact  throws  doubt  upon 
the  assumption  of  a  plane  surface  of  rupture  for 
all  cases  where  the  direction  of  the  thrust  on  the 
vertical  plane  does  not  act  parallel  to  the  upper 
surface. 

It  appears  reasonable  to  suppose,  if  the  line 
through  the  centres  of  pressure  on  all  sections 
of  a  retaining-wall  passes  through  their  centres 
of  gravity,  that  no  rotation  of  the  wall  occurs; 
further,  if  it  was  possible  for  the  masonry  and 


65 


earth  or  rock  backing  to  settle  together  the  same 
amount,  —  the  backing,  say,  having  been  carefully 
deposited  in  horizontal  layers,  —  then,  even  fora 
level-topped  bank,  the  maximum  thrust,  as  given 
by  Rankine's  formula,  will  be  exerted,  and  there 
will  be  no  friction  at  the  back  of  the  wall  to  change 
the  usual  direction  of  the  earth-thrust  in  a  large 
bank.  If  the  wall  has  not  the  stability,  or  the 
settling  is  not  as  assumed,  the  top  will  move  over, 
friction  at  the  back  of  the  wall  is  exerted,  and  the 
horizontal  thrust  becomes  smaller  than  before, 
corresponding  to  a  different  prism  of  thrust,  as 
we  ascertain  by  the  construction  of  Fig.  4,  for 
the  two  cases  of  E  horizontal  and  E  inclined 
downwards.  The  excess  of  the  horizontal  thrust 
in  one  case  over  that  in  the  other  must  necessarily 
be  resisted  by  the  ground-surface,  on  which  the 
filling  rests  by  friction,  which  it  is  generally 
capable  of  doing.  If  not,  then  the  Rankine 
thrust  will  be  exerted.  Similarly,  if  we  consider 
any  road  embankment,  whose  sides  slope  at  the 
angle  of  repose,  the  horizontal  thrust  on  some 
longitudinal  plane  in  the  interior  must  be  finally 
resisted  by  the  ground  to  one  side  under  the 
embankment.  If,  however,  the  weight  of  earth 
above,  multiplied  by  the  co-efficient  of  friction  of 
earth  on  ground-surface,  is  less  than  the  horizontal 
thrust,  the  earth  must  slide,  and  the  slope  become 
flatter,  until  equilibrium  obtains  from  a  less 
horizontal  thrust.  Scheffler  computes  for  an 
embankment  of  triangular  section  where  <j>  =  45°, 


66 


and  the  angle  of  friction  on  the  ground-surface  is 
only  5°,  that  the  slope  of  the  embankment  would 
change  to  32°  15'.  For  the  ground-friction  angle 
—  7°  7'  20"  there  would  be  exact  equilibrium;  so 
that,  generally,  there  need  be  no  fear  from  spreading 
of  embankments  due  to  this  cause,  as  the  amount 
of  friction  required  is  very  small. 

34.  We  have  now  given  methods  for 
finding  the  thrust  against  a  retaining-wall, 
which  simply  resists  this  active  thrust  of  the 
earth,  for  the  usual  cases  of  a  surcharged 
wall  and  earth-level  at  the  top,  to  which 
may  be  added  the  case  of  earth  sloping 
downwards  from  the  top  of  the  wall  to  the 
rear,  for  which  the  construction  is  evident. 
It  now  remains  to  find  the  passive  resistance 
of  the  earth  to  sliding  up  some  inclined  plane 
due  to  an  active  thrust  of  the  wall  from  left 
to  right  (Fig.  4),  caused  by  water,  earth, 
or  any  other  agency  acting  against  the  wall 
on  the  left.  Now  (Art.  22)  we  lay  off  the 
angle  bgs  (Fig.  4)  above  bg,  and  then,  from 
the  new  position  of  s,  lay  off  arcs  ssr  ss2, 
.  .  .  ,  below  s  equal  to  gav  ga2,  .  .  .  ,  as 
before,  giving  the  direction  of  gsr  gs.2,  .  .  .  , 
inclined  at  anle  above  the  normal  to  the 


67 


corresponding  planes  ^41,  ^42,  .  .  .  The 
construction  then  proceeds  as  before,  only 
it  is  now  the  least  of  the  resistances,  </c,  that 
represents  the  passive  resistance  of  the 
earth  to  sliding  up  the  plane  of  rupture 
corresponding ;  for  any  increase  over  this 
causes  the  thrust  on  some  planes  to  make 
greater  angles  than  (f)  with  the  normals,  as 
is  easily  shown.  Let  us  call  N  the  com- 
ponent normal  to  AC  of  the  resistance  and 
suppose  a  slight  movement  of  the  wall  hori- 
zontally to  the  right;  then  since  the  earth 
moves  upwards  along  the  plane  of  rupture 
and  the  plane  AC,  the  friction  of  the  earth 
along  AC,  N  tan  </>,  acts  upwards  and  the 
resistance  of  the  wall  downwards. 

The  thrust  E  is  now  inclined  at  the  angle 
(f>  above  the  normal  to  A C  and  nob  below  s,s 
formerly.  The  active  thrust  is  of  course  the 
only  one  exerted,  unless  the  wall  tends  to  slide, 
so  that  the  consideration  of  the  passive  resist- 
ance is  of  small  practical  value.  In  case  of  a 
heavy  structure  resting  on  a  foundation,  we 
can  replace  the  total  weight  by  that  of  earth, 
and  estimate  the  active  thrust  exerted  pgainst 


68 


a  vertical  plane  just  below  the  foundation, 
for  the  full  weight  of  the  supposed  earth, 
by  the  method  to  be  given  in  the  next 
article.  The  earth  to  one  side  of  this 
vertical  plane  can  be  conceived  to  exert  a 
passive  thrust,  which  may  be  estimated  as 
explained,  and  should  exceed  the  active 
thrust  for  a  stable  foundation.  This 
method,  though,  of  estimating  the  stability 
of  a  foundation,  while  doubtless  on  the 
safe  side,  is  otherwise  illusory,  as  any  one 
who  has  seen  a  heavy  locomotive  move  at 
great  speed  along  a  narrow  embankment 
must  admit.  The  mass,  by  its  friction, 
rapidly  and  safely  transmits  and  distributes 
the  weight  over  the  ground,  without  exerting 
any  horizontal  thrust  at  the  side  slopes, 
which  are  perfectly  stable. 

35.  Underground  Pressures.  —  To  find 
the  unit  pressure  at  a  depth  x  below  the 
surface  of  a  large  mass  of  earth,  level  at 
top,  of  indefinite  extent,  and  resting  upon 
a  uniformly  compressible  foundation,  every 
where  at  the  same  depth  (see  Art.  9),  we 
proceed  as  follows :  Let  Fig.  6  represent  a 
slice  of  the  earth  contained  between  two 


69 


vertical  planes  one  unit  apart,  and  bounded 
on  one  side  by  the  horizontal  plane  (K7,  at 
a  depth  x  below  the  surface,  on  the  left 
by  the  vertical  plane  ^10,  whose  depth  is 


Ax,  and  below  by  the  plane  AC ;  the 
planes  AQ,  0(7,  and  AC  being  supposed 
perpendicular  to  the  plane  of  the  paper. 
Let  the  length  -40  =  Ait1,  and  the  length 
OU  =  w.Ax.  The  plane  AC  will  be  con- 


70 


sidered  to  take  successively  the  positions 
.41,  A2,  .  .  .  ;  so  that  if  we  divide 
AO  =  Ao?  into  ten  equal  parts,  as  shown, 
and  lay  off  similar  equal  parts  along  (H7, 
as  AC  varies  in  position,  n  will  take  the 
successive  values  0.1,  0.2,  .  .  .  Calling 
e  the  weight  per  cubic  foot  of  earth,  the 
weight  of  the  prism  of  earth  resting  verti- 
cally over  (X7  is  represented  generally  by 
e.x.n.  Asc,  which,  being  directly  proportional 
to  n,  we  can  lay  off  on  the  vertical  OA  the 
lengths  01,  02,  .  .  .  ,  to  represent  the  suc- 
cessive values  of  w,  or  the  vertical  loads 
sustained  by  the  horizontal  bases  01,  02, 
.  .  .  ,  of  the  successive  prisms  considered. 
When  the  length  Aa?  is  very  small,  we  can 
neglect  the  weight  of  the  small  prism  of 
thrust,  .40(7,  in  comparison  with  the  weight 
of  the  vertical  prism  above  it,  without 
appreciable  error,  and  ultimately  find  the 
position  of  the  plane  AC,  which  gives  the 
true  thrust  against  Ati,  by  previous  methods. 
Thus,  draw  the  quadrants  shown  with  A 
and  0  as  centres,  and  AO  as  a  radius  ;  note 
the  intersections  av  a2,  .  .  .  ,  of  the  lines 
,41,  A2,  .  .  .  ,  with  the  arc  OD ;  next, 


71 


construct  angle  Cos  =  <f>  the  angle  of 
repose  of  the  earth,  and  arcs  ssl  =  0«p 
ss^  =  0«.2,  .  .  .  ;  so  that  each  of  the  lines 
Osr  Os2,  .  .  .  ,  next  drawn,  make  the  angle 
(f)  with  the  normals  to  the  corresponding 
planes  Al,  A2,  .  .  .  ,  and  thus  represent 
the  direction  of  the  resistances  offered  by 
these  planes  in  turn  regarded  as  planes  of 
rupture.  On  drawing  horizontals  through 
the  points  of  division  1,  2,  .  .  .  ,  on  AO  to 
intersection  1",  2",  .  .  .  ,  with  the  cor- 
responding directions  Osv  Os2,  .  .  .  ,  we 
note,  that,  if  the  thrust  on  .40  is  taken  as 
horizontal  (Art.  9),  the  lines  11",  22", 
.  .  .  ,  represent  the  horizontal  thrusts 
caused  by  the  weights  resting  on  the  suc- 
cessive prisms  .401,  .402,  .  .  .  ,  treated  as 
successive  wedges  of  rupture.  The  greatest 
of  these  7T77  represents  the  actual  thrust  on 
AO  ;  for  if  we  assert  that  any  other,  as  4I77, 
represents  the  actual  thrust,  to  get  the 
corresponding  thrusts  on  all  the  planes  .41, 
.42,  .  .  .  ,  in  direction  and  amount,  we 
must  lay  off  a  length  equal  to  W  along 
each  of  the  horizontals  11",  22",  .  .  .  , 
produced  if  necessary,  and  through  the 


72 


extremities  draw  lines  to  0,  which  thus 
represent  in  amount  and  direction  the 
thrusts  on  the  corresponding  planes.  But 
since  1477  is  less  than  7T7,  this  construction 
will  give  a  thrust  on  the  plane  ^47,  lying 
below  the  position  07",  and  thus  making  a 
greater  angle  than  c£  with  its  normal,  which 
is  inconsistent  with  the  laws  of  stability  of 
a  granular  mass.  Hence,  any  other  thrust 
than  the  maximum,  as  given  by  the  above 
construction,  is  impossible. 

The  length  of  77^  to  scale  is  0.52,  which 
we  must  now  multiply  by  ex&x  to  get  the 
total  horizontal  thrust  on  the  plane  JU  in 
pounds.  On  dividing  this  thrust  (0.52 
ex&x}  by  the  area  pressed  =  1  X  A#,  we 
get  the  unit  pressure  on  a  vertical  plane 
at  a  depth  x  below  the  surface  equal  to 
0.52e.x,  which  is  called  "the  intensity  of 
pressure,"  at  a  depth  x.  As  we  neglected 
the  weight  of  the  prism  AO C.  we  must 
conceive  A#  to  diminish  indefinitely,  so  that 
the  error  tends  indefinitely  towards  zero, 
and  the  approximate  intensity  of  pressure 
on  JI)  =  A#  approaches  indefinitely  that 
at  the  point  0. 


73 


By  analysis  we  shall  show  hereafter  that 
the  plane  of  rupture,  Al  in  this  case,  bisects 
the  angle  between  the  natural  slope  and  the 
vertical. 

In  this  case  we  have  taken  0  =  18°  26',  and  the 
resulting  intensity  (0.52ex)  is  found  to  be  exactly 

that  given  by  the  usual  formula,  ex  tan2  (45°  —  -  Y 

The  intensity  at  any  point  of  a  vertical  plane  thus 
varies  directly  with  x.  The  total  amount  on  a 
vertical  plane  of  depth  x  from  the  surface  is  then 

rCx2 
Cxdx  = (where  C  =  0.52e  in  the  present 
z 

case),  and  its  resultant  is  at  a  depth  z  equal  to  the 
limit  of  the  sum  of  the  moments  of  the  pressures 
(Cxdx)  on  the  elementary  areas  dx  x  1,  taken  about 
the  top  surface,  divided  by  the  total  pressure,  or 

,  =f  «*  +  f  =  f. 

Also,  £~  =  ef  X  0.52x  =  ^  X  line  represent- 

Z  Z  £i 

ing  thrust,  if  old  construction  is  used.  These 
are  precisely  the  conclusions  derived  from  previous 
constructions. 

In  case  the*  top  surface  is  sloping,  a 
similar  construction  applies,  only  ~QC  must 
now  be  drawn  parallel  to  the  top  slope,  and 
the  pressure  on  OA  must  be  assumed  to  act 


74 

parallel  to  this  direction.  The  construction 
is  similar  to  that  given  for  Fig.  5  (on 
neglecting  the  weight  of  the  wedge  of 
thrust  as  above),  either  to  the  right  or  left 
of  the  vertical  Ag,  only  as  the  weight  of 
the  prisms  vertically  above  01,  02, 
(Fig.  5)  is  now  represented  by  ex 
cos  i  (where  i  is  the  inclination  of  the  top 
slope  to  the  horizontalVwe  must  multiply 
the  length  of  the  line  ~cc'  (Fig.  5)  to  scale, 
by  ex  cos  i,  to  get  the  intensity  of  the 
pressure  at  the  depth  x,  since  the  lengths  n 
alone  were  laid  off  to  represent  the  loads 
99*i  99%  -  -  •  i  as  'lu  Fig.  6,  and  the  resulting 
thrust  cc'  must  now  be  magnified  ex  A#  cos 
i  times  to  get  the  thrust  in  pounds  on  the 
plane  A#  X  1.  As  A#  approaches  zero 
indefinitely,  the  approximate  intensity 

ex   AX  cos  i  '—- 

-  cc  ,  on  the  area   &x  X   1,  ap- 

A3? 

proaches  that  at  the  depth  x  (—ex  cos  i.  cc') 
as  near  as  we  please.  It  must  be  observed 
that  ZO  in  Fig.  5  must  be  taken  equal  to 
unity  in  this  construction,  and  the  same 
scale  used  in  laying  off  the  distances  along 
the  top  slope  0  —  10. 


75 


36.  If  the  earth  to  the  right  of  AO,  in 
Fig.  6,  does  not  experience  the  similar 
active  thrust  of  earth  to  the  left  of  J.O,  but 
only  the  passive  resistance  of  a  tunnel 
lining,  etc.,  of  an  underground  structure, 
the  conditions  are  changed  if  this  lining 
gives  in  consequence  of  its  elasticity ;  for 
the  wedge  of  thrust,  J.OC,  cannot  move 
to  the  left  without  developing  friction  along 
the  surface  0(7,  therefore  the  pressure  on 
this  surface  must  no  longer  be  taken  as. 
vertical,  but  as  inclined  at  a  direction 
0 — 1C',  making  an  angle  <£  with  the  vertical 
(Fig.  6).  The  load  on  any  supposed  wedge 
of_  thrust,  as  .A 04,  is  now  represented  by 
04',  the  thrust  on  H)  by  ^Vand  the 
pressure  on  the  plane  A4  by  04".  The 
greatest  of  the  lines,  I'l",  2'2",  .  .  .  ,  now 
represents  the  true  thrust,  and  it  is  readily 
found  to  be  4/4//  =  .33  to  scale  j  so  that  the 
intensity  of  the  thrust  on  a  square  foot  at 
the  depth  x  is  now  0.33e.r,  or  one-third  the 
intensity  on  the  horizontal  plane  0(7.  Mr. 
Baker  ("Science  Series,'7  No.  56)  found 
for  a  heading,  driven  for  the  Campden-Hill 
Tunnel,  at  a  depth  of  44  feet  from  the 


76 

•surface, — the  angle  of  repose  of  the  over- 
lying clay,  sand,  and  ballast,  heavily 
charged  with  water,  being  only  18°  26'  as 
.assumed  above, — that  the  relative  deflec- 
tions of  the  timbering  in  the  roof  and  sides 
indicated  that  the  vertical  and  horizontal 
intensities  of  pressure  were  in  the  ratio  of 
3.5  to  1,  which  is  very  near  what  we 
obtain  by  the  last  construction.  The  first 
construction  indicates  a  ratio  of  only  2 
iol. 

In  most  cases,  a  portion  of  the  weight  of 
"the  earth  abovo  the  tunnel  is  transferred 
"to  the  sides  (Art.  9),  though  here  it  was 
thought  that  "the  full  weight  of  the 
ground  took  effect  upon  the  settings." 

We  have  now  carefully  examined  the 
conditions  of  interior  equilibrium  of  a  mass 
•of  earth,  and  ascertained  the  thrusts  ex- 
erted, whether  in  the  interior  or  against  a 
retaining- wall  j  and  we  see  that  the  graphi- 
cal method  is  capable  of  handling,  with 
«qual  ease,  any  case  that  ordinarily  pre- 
sents itself.  The  results,  of  course,  agree 
with  the  analytical  method,  founded  on  the 
same  hypotheses  j  but  as  it  is  often  more 


77 

convenient  to  calculate  the  thrust,  even 
when  a  graphical  method  is  afterwards 
used  for  testing  the  stability  of  the  wall, 
we  shall  now  proceed  to  deduce  formulas 
for  evaluating  it. 


78 


CHAPTER  III. 

THEORY   OF   RETAINING- WALLS. 

Analytical  Method. 

37.  As  in  the  preceding  chapter,  we 
shall  assume  a  plane  surface  of  rapture, 
and  regard  the  mass  as  subject  only  to  the 
laws  of  gravity  and  friction  al  stability 
stated  in  Art.  21. 

In  Fig.  7,  let  AFPQEC  represent  a 
cross-section  of  the  earth-filling,  taken  at 
right  angles  to  the  inner  face  of  the  wall 
A.F.  We  shall  consider  the  conditions  of 
equilibrium  of  a  prism  of  this  earth  con- 
tained between  two  parallel  planes,  per- 
pendicular to  the  inner  face  of  the  wall,, 
and  one  unit  apart,  regarding  the  wall 
AF  as  resisting  the  tendency  of  the  earth 
to  slide  down  some  plane,  as  AC,  passing 
through  its  inner  toe. 

Call  G  the  weight  of  the  prism  of  earth 


79 


in  pounds,  directed  vertically  j 
E,  the  earth-  thrust  against  the  wall  AFT 
directed  at  an  angle  <f>f  of  friction  of  earth 
on  wall  when  <f>'  <  <£,  or  of  </>  when  <J>'  >  ^ 


below  the  normal  to  the  inner  face  of  the 
wall  (Art.  7);  and  $  the  reaction  of  the 
plane  AC,  inclined  at  an  angle  <£  (the  angle 
of  repose  of  earth)  below  the  correspond- 
ing normal,  since  the  prism  is  supposed  to 
be  on  the  point  of  moving  down  the  plane 


80 

A  C.  These  three  forces  are  in  equilibrium 
when  E  and  S  act  towards  0  and  G  acts 
downwards. 

Call  the  angle  that  AC  makes  with  the 
horizontal  y,  and  the  angle  FAC,  ft.  On 
drawing  the  parallelogram  of  forces  as 
shown,  we  have,  since  E  and  6r  are  pro- 
portional to  the  sines  of  the  opposite 
angles  in  the  triangle  ONL, 

E  _  sin  ONL 
~G  ~~  sin  NLO 

It  is  easily  seen  from  the  figure  that  ONL  — 
y  —  t,  and  that  NLO  =  <£  -f-  ft  +  <£'  ; 
hence  the  above  general  relation  becomes, 


a     ~   sin  (<£  +     '  +  ft) 

Now,  if  we  conceive  the  plane  A  C,  always 
passing  through  the  point  A,  to  vary  its 
position,  that  value  of  E,  corresponding  to 
the  greatest  value  obtained  by  the  con- 
struction above,  is  the  thrust  actually  ex- 
erted against  the  wall  ;  for,  if  A  C  is  the 
plane  of  rupture  corresponding  to  this 
greatest  trial  thrust,  any  less  value  of  the 


81 


resistance  of  the  wall  E  will  cause  8  to 
make  an  angle  greater  than  <£  with  the 
normal  to  AC,  which  (Art.  21)  is  inconsist- 
ent with  the  law  of  stability  of  a  granular 
mass  (also  see  Art.  25) :  hence  the  least 
thrust  consistent  with  equilibrium  corre- 
sponds to  the  greatest  value  of  E  thus  ob- 
tained 5  and  this  is  the  actual  active  thrust 
exerted  against  the  wall,  when  the  wall 
simply  resists  the  tendency  to  overturning 
or  sliding  on  its  base,  caused  by  the  ten- 
dency of  the  prism  of  rupture  to  descend. 
If  there  is  a  thrust  exerted  on  the  wall  to- 
wards the  earth,  from  any  external  force' 
acting  on  the  left  of  the  wall  j  from  left  to 
right  5  then,  if  this  be  supposed  to  increase 
gradually,  the  act  ire  thrust  of  the  earth  on 
the  right  is  first  overcome  j  then,  as  the  ex- 
ternal force  increases,  the  directions  of  S, 
on  all  planes  as  ACy  approach  the  normals 
to  those  planes,  pass  them,  and  finally  the 
full  passive  resistance  of  some  prism  of  earth 
to  sliding  upwards  along  its  base  is  brought 
into  play.  The  greatest  force  E,  as  regards 
sliding  up  the  base  of  some  prism,  which 
can  be  exerted  is  that  corresponding  to  the 


least  of  the  trial  forces,  E,  obtained  by 
supposing  the  position  of  the  plane  A  C 
to  vary,  for  S  lying  above  the  normal  to 
A  C  at  an  angle  <£  for  each  plane ;  for  if  we 
suppose  A  C  to  represent  the  corresponding 
plane  of  rupture,  if  the  external  force, 
equal  to  E,  and  acting  from  left  to  right,  is 
increased,  it  necessarily  causes  the  direc- 
tion of  S  to  make  a  greater  angle  than  <£ 
with  the  corresponding  normal,  which  is 
inconsistent  with  equilibrium  (Art.  21). 

In  this  chapter  we  shall  only  consider 
the  passive  resistance  of  the  wall  to  over- 
turning or  sliding  caused  by  the  active 
thrust  of  the  earth  tending  to  descend, 
which  is  all  that  is  required  in  estimating 
the  stability  of  retaining-walls. 

38.  We  shall  now  express  the  value  of 
G  for  the  earth-profile  shown  in  Fig.  8 
taken  to  represent  the  general  case,  and 
proceed  to  find  the  maximum  value  of  E, 
for  different  trial-planes,  which  represents 
the  actual  thrust  exerted  against  a  stable 
wall.  We  shall  suppose  the  true  plane  of 
rupture  to  intersect  the  part  EY  of  the 
profile ;  the  line  EY  is  then  produced  to 


83 


B,  so  that  the  area  of  the  triangle  ABC 
is  equal  to  that  of  the  polygon  AFPQRC, 
which  can  be  effected  by  ordinary  geomet- 
rical means.  The  point  B  therefore  does 
not  change,  as  we  suppose  the  position  of 
C  to  vary  between  E  and  Y. 


.  s 


Let  us  drop  the  perpendicular  AT  from. 
A  upon  B  Y,  and  designating  by  e  the 
weight  per  cubic  foot  of  earth,  we  have 

a  =  &.AT.BC. 

For  future  convenience  we  have  desig- 
nated, in  Fig.  8,  the  angle  that  A  C  makes 
with  the  vertical  &?,  and  the  angle  that  the 
inner  face  of  the  wall  AF  makes  with  the 


84 

vertical  a;  so  that  the  angle  /3  of  (1)  is 
now  replaced  by  (co  -f-  «')  if  the  wall  leans 
forward,  or  by  (GO  —  a)  if  the  wall  leans 
backwards. 

In  Fig.  8,  let  us  draw  the  line  CT,  mak- 
ing the  angle  ACI  =  (<j>  -f-  <J>'  -f  /j)  = 
(<£-]-  <//  +  oo  -f~  «')  to  intersection  J7  with 
the  line  of  natural  slope  AD  through  A. 
If  the  wall  leans  backwards, 

ACI—  (<£  +  <£'  -f  <»  —  <*). 

Since  the  angle  (y  —  cp]  =:  CAT,  we  can 
replace  the  ratio  of  the  sines  in  (1)  by  that 
of  the  sides  opposite  in  the  triangle  A  CI, 
or  of  CI  to  AI;  so  that,  substituting  the 
above  value  of  G,  we  can  write  (1)  in  the 
following  form  :  — 


E=  be.AT.BC.         .  .  .  (2). 
AI 

On  drawing  SO  parallel  to  CI  to  inter- 
section 0  with  AD,  we  have,  from  this 
relation  and  the  similar  triangles,  BOD 
and  CID. 


85 
which  substituted  in  (2)  gives, 


E  =  .    OLID        _  (3 

V       QD2       I       AI      • 

The  terms  in  the  (  )  remain  constant  as 
we  vary  the  position  of  A  C.  For  brevity, 
call  J./=  x,  AD  —  a,  AO  =  b  ;  then  we 
can  write  the  variable  term, 

OLID  _  (x  —  l)  (a-x}_a  ,  ^       ab      x 
AI  x  x 

which  is  a  maximum  for  x  —  ^/ab^  as  we 
find  by  placing  its  first  derivative  equal  to 
zero.  This  value  of  x  substituted  in  the 
variable  term  gives, 


so  that  the  actual  thrust  E  on  the  wall  can 
be  written,  — 


K-l...- 
\       OD*       /          a 

Now,  drawing  the  perpendiculars  BN  and 
CI1  from  B  and  C  upon  AD,  we  observe 
that  since  the  angle  A  CH  =  co  -j-  <£  (A  C 
makes  the  angle  GO  with  a  vertical  at  C, 


86 

and  CH  makes  the  angle  <£  with  this  same 
vertical,  since  the  sides  are  respectively  per- 
pendicular to  those  of  the  angle  DAJ~<$>\ 
and  the  whole  angle  A  C!=(GJ +<}>+<}> '+a'), 
it  follows  that  .the  angle  HCI  =  NBO  — 
(cf>'  -\-  a)  as  marked,  if  the  wall  leans  for- 
wards; otherwise  HCI=NBO=(<j>'—a)y 
since  .ACT is  then  equal  to  (Go-\-<t>+<t>' — a), 
as  previously  observed. 

To  reduce  (4)  to  a  simpler  form,  we 
remark  that  AT.BD  represents  double  the 
area  of  the  triangle  ABD,  and  can  be  re- 
placed by  AD.BN  =  AD.BO  cos  OBN ; 
which  gives 
ATBD.BO 


=  acos  OBNl--): 

oif  \OD)  ' 

=  ie.  cos  OBN  (- 


(a  —  y)2  .  .  .  (5). 

Now,  from  similar  triangles,  BOD,  CID, 

TtO        C*T 

we  have  ——  =  — ,  which,  substituted  in 

the  above  expression,  we  have,  noting  that 
(a  —  4/ab)  =  (a  —  x)  —  ZZ),  the  very  sim- 
ple formula, 


87 


E=  \e.  cos  (<£'  +  a)  CI-  .  .  .  (6). 

It  is  to  be  remarked,  that,  if  the  wall  leans 
backwards,  cos  (<£'  -}-  a)  is  to  be  replaced 
in  this  formula  by  cos  (<£'  —  ex)  ;  further,  if 
we  lay  off  IL  =  1C  on  the  line  IAf  and 
draw  a  line  from  L  to  C,  the  thrust  E  is 
exactly  represented  by  the  area  of  the 
triangle  ICL  multiplied  by  e,  the  weight 
per  cubic  foot  of  the  earth. 

39.  This  simple  conclusion  has  been 
previously  reached,  in  an  entirely  different 
manner,  by  Weyrauch  (see  "Van  Nostrand's 
Magazine"  for  April,  1880,  p.  270),  who 
states  that  Rebhahn  in  1871  found  a  similar 
result,  assuming,  however,  that  <f>'  =  0.  or 
<f>'  —  (f>  (for  the  special  cases  of  earth-level 
at  top,  or  sloping  at  the  angle  of  repose,  I 
infer). 

Recurring  now  to  the  fact,  that  for  the 
true  plane  of  rupture  we  found 
x  =  A  I  = 


and  that  angle  NBO  =(<£'-far)  or  ($'—  a), 
according  as  the  wall  leans  forwards  or 
backwards,  we  have  the  following  simple 
construction  to  find  the  plane  of  rupture 


88 


and  earth-thrust  E,  as  given  by  Weyrauch 
in  1878,  for  a  uniform  slope  and  wall  lean- 
ing forward. 

Having  found  the  point  B  on  the  pro- 
longation of  the  line  RY,  which  it  is  thought 
will  be  intersected  by  the  plane  of  rupture, 
so  that  area  ABE  rr  area  AFPQR,  we 
next  draw  J50,  making  the  angle  NBO 
with  the  normal  to  the  line  of  natural  slope 
AD,  equal  to  (<j>'  -f-  a)  or  (</>'  —  a),  accord- 
ing as  the  inner  face  of  the  wall  lies  to  the 
left  or  to  the  right  of  the  vertical  through 
A  (replace  <£'by  <£  whenever  <£'><£)  j  then 
erect  a  perpendicular  at  0  to  AD  to  inter- 
section M,  with  the  semicircle  described 
upon  AD  as  a  diameter,  and  lay  off  AI  = 


chord  AM,  since  AI  —  y'A  O.AD  >  next, 
draw  1C  parallel  to  OB  to  intersection  C 
with  the  top  slope,  whence  A  C  will  be  the 
plane  of  rupture  if  the  point  C  falls  upon 
JtY  as  assumed  j  otherwise  another  plane, 
as  YZ,  will  have  to  be  assumed  as  con- 
taining the  point  (7,  and  the  construction 
effected  as  before. 

Having  found  C  in  this  manner,  E  can 
be  computed  from  (1),  since  Cr  =  b  A  T.BC 


89 


is  now  known :  or  by  measuring  CI  to 
scale,  E  can  be  found  directly  from  (6) 

This  graphical  construction  is  more 
rapid  and  accurate  in  working  than  the 
methods  of  the  preceding  chapter,  and  is 
superior  to  Poncelet's  construction,  in 
taking  less  space  to  effect. 

In  surcharged  walls,  the  point  B  will 
generally  lie  to  the  right  of  A  F.  Thus,  in 
Fig.  4  the  upper  line  26  is  extended  to  the 
left ;  from  0  a  line  is  then  drawn  parallel 
to  A2  to  intersection  0'  with  the  line  26 
extended.  The  point  0'  thus  found  corre- 
sponds to  the  point  B  of  Fig.  8. 

40.  The  construction  is  true  whether  the 
earth-surface  slopes  upwards  or  down- 
wards from  the  top  of  the  wall. 

In  the  latter  case,  if  the  surface,  say  BD, 
falls  upon  the  line  BO,  the  construction 
fails ;  but  a  formula  given  farther  on  gives 
the  value  of  E. 

If  the  surface  BD  falls  below  BO,  it  is 
easily  seen,  on  drawing  a  figure,  that  all  the 
previous  equations  hold,  and  we  reach  the 
same  conclusion  as  before,  A  J=  ^/AD.A  0  j 
only  as  AO  now  is  larger  than  AD,  the 


90 


semicircle  must  be  described  upon  A  0  as 
a  diameter,  and  the  perpendicular  to  the 
point  M  erected  at  D  ;  or  A  Jean  be  calcu- 
lated if  preferred.  If  the  points  0,  J, 
and  D  are  near  together,  it  will  be  best  to 

compute  BC  from  BC  —  BD.--,  since 

the  terms  in  the  right  member  can  be 
measured  to  scale. 

4i.  Position  of  the  Limiting  Plane. — In 
Fig.  9;  let  BD  represent  the  earth-surface, 
uniformly  sloping  at  the  angle  i  to  the 
horizontal,  of  an  unlimited  mass  of  earth 
(Art.  9),  in  which  the  pressure  on  a  verti- 
cal plane,  AB,  can  be  taken  as  parallel  to 
the  surface  BD.  Let  AD  represent  the 
line  of  natural  slope ;  it  is  required  to  find 
the  position  of  the  plane  of  rupture  A  C, 
corresponding  to  the  thrust  E,  acting 
above  the  horizontal  at  the  angle  i,  and  of 
course  balancing  the  opposed  thrust  of 
the  earth  to  the  left  of  AB. 

On  referring  to  Fig.  7,  it  is  seen  that 
equation  (1)  holds  on  replacing  the  de- 
nominator of  the  right  member  by  sin 
i)-  Therefore,  in  Fig.  8,  the  angle 


91 


ACI  must  now  be  laid  off  equal  to 
(fi+<f> — «),  whence  the  line  CI  falls  below 
CH,  and  BO  below  BN,  both  being  in- 
clined to  these  normals  at  the  same  angle; 
^-j-«=i-fO=ti 

With  this  exception,  the  above  demon- 
stration holds  throughout,  and  we  reach 


the  following  construction  to  find  the  point 
C.  From  B  draw  BO,  making  the  angle  i 
below  the  normal  BNio  AD,  or  preferably 
making  the  angle  (<f>  —  i)  with  the  vertical 
AB,  to  intersection  0  with  AD.  From  0 
draw  OM perpendicular  to  AD  to  intersec- 
tion Mj  with  the  semicircle  described  upon 


92 


AD  as  a  diameter ;  lay  off  AI  along  AD, 
equal  to  chord  AM,  and  from  I  draw  a 
parallel  to  BO  to  intersection  C  with  the 
top  slope  BD.  The  plane  A  Cis  the  plane 
of  rupture,  or  the  limiting  plane  of  Art.  28, 
which  see. 

If  the  inner  face  of  the  wall  lies  below 
AC,  then  (Art.  28)  the  thrust=ie.  cos  i.  C/2 
on  AB  is  computed,  and,  regarded  as 
acting  parallel  to  BD,  from  left  to  right, 
is  combined  with  the  weight  of  the  earth 
and  wall  to  the  right  of  AB  to  find  the 
true  resultant  on  the  base  of  the  wall. 

If  the  wall  lies  between  AB  and  A  C,  the 
constructions  of  Arts.  37  and  38  are  used. 

To  be  as  accurate  as  possible  in  these,  as 
in  all  constructions,  true  straight  edges  on 
both  ruler  and  triangle  are  imperative. 
Lay  off  all  angles,  including  right  angles, 
by  aid  of  a  beam  compass  to  a  large 
radius,  say  ten  inches,  using  a  table  of 
chords  (except  for  the  right  an^le)  and  an 
accurate  linear  scale.  With  all  care,  the 
angles  BAG  thus  found  can  scarcely  be 
counted  on  to  nearer  than  ten  minutes, 
which,  however,  is  sufficiently  accurate. 


93 


In  the  table  below  will  be  found,  for 
various  inclinations  i,  the  values  of  the 
angle  BAG  that  the  limiting  plane  makes 
with  the  vertical  ;  also  the  co-efficient  K 
(see  Art.  42),  or  the  thrust  on  AB—^e 
cos  i  6T2,  when  AB  and  e  are  both  taken 
as  unity,  made  out  for  earth  which  naturally 
takes  a  slope  of  one  and  a  half  to  one,  or 
whose  angle  of  repose  is  33°  42'. 

The  value  of  K  agrees  fairly  well  with 
calculation,  the  last  figure  not  differing 
more  than  one  or  two,  at  the  outside,  from 
the  results  of  Art.  47. 

From  the  construction  we  see  that  as  i  ap- 
proaches </>  indefinitely,  B A  C  tends  to  zero 
and  E  approaches  the  limit  \e  cos  <j>.  AB2,  as 
given  by  analysis.  The  increase  of  thrust 
is  very  rapid  from  *=30°  to  t=^  =  33°  42'. 


i 

0°   J5° 

1CC 

15° 

20° 

25° 

26°34 

30° 

33°  42' 

BAG 

28°09'  26° 

24° 

21°  50' 

19°  10' 

16° 

14°  40' 

11°  10' 

0° 

E 

.143  .145 

.149 

.,57 

.172 

.194 

.207 

.244 

.416 

94 

42.  Uniform  Top  Slope;  Formula  for 
Earth-thrust. — When  the  upper  surface  of 
the  earth  slopes  uniformly  at  the  angle  i  to 
the  horizontal,  it  is  easy  to  deduce  from 
what  precedes  a  general  formula  for  the 
thrust  exerted  by  it.  Fig.  10  represents  a 


Fig.  10 


retaining-wall  leaning  towards  the  earth. 
We  shall  first  deduce  a  formula  for  this 
case,  when  it  will  be  observed,  as  we  pro- 
ceed, that  the  same  formula  holds,  when 
the  wall  leans  forward/ on  simply  changing 
a  to  (  —  a). 

In  this  case,  we  note  from  Fig.  10  the 
following  values  for  the  angles. — 


95 

NBO  =  V  —  a, 
ABO  =  <£  +  <£', 
AOB  —  90  —  (<£'  —  a), 
ADB  =  <j>  —  i, 


Finally,  designate  by  Z  the  length  .AJ?  from 
the  inner  toe  to  where  the  inner  face  of  the 
wall  intersects  the  top  slope,  and  by  h  its 
corresponding  vertical  projection. 

From  formula  (5)  we  deduce,    remem- 
bering that  OD  =  (a—l), 


E  =  ±e.  BO*       ~     a.  ™s  OBN.  .  .  (7). 
La  —  b  J 

We  can  now  write  the  [  ]  as  follows  :  — 

It 

a—  Vab  __  1  —  \a  1 


a-b 


Placew  = 


£ 
Na 


P.  to  find  its  value  in  .terms 

N« 

of  the  functions  of  known  angles,  we  have 
from  the  triangles  AOB  and  ABD  by  the 
law  of  sines, 


96 

AO  __  sin  (<f>  +  $')    AB  _  sin  (<f>  —  i) 
AB        cos  (V  —  a)'  AD  ~  cos  (or  +  i)' 

On  multiplying  these  two  equations  to- 
gether, and  extracting  the  square  root,  we 
find, 


_      ,J     —     /  sn  (<>  +  <)  sn  (0- 
~  \A£>      \  cos  (^-a)  cos  ' 


Again,  from  the  triangle  BOA,  we  have, 


cos  (<f>f  —  a) 

Substituting  these  values  in  (7),  and  putting 
cos  OBN=  cos  (<£'  —  a)  for  this  case,  and 
we  have  finally, 


n  +  1     /   2  cos  (<£'  —  <ar) 
Or,  since  h  =  /  cos  <*7  we  likewise  have, 


^)\2  ^2  no) 

al  2  cos    <x  —  a    ' 


cos  al  2  cos  (<^x  —  a) 

If  we  term  the  co-efficient  of  eh2  in  (10), 
j  we  can  write  this  formula, 
E  =  Keh2  .  .  .  (11) 


97 


in  which,  for  walls  leaning  backwards,  as  in 
Fig.  10, 


cos  a)  2  cos  (<£'  —  a) 

where  n  has  the  value  given  in  (8). 

For  ivalls  leaning  forwards,  we  easily 
note  the  changes  in  the  angles  used,  and 
can  verify  that  formula  (11)  obtains;  but 
now, 

K  —  /cos  (<ft  —  ^)\2 1 

V(M-f-l)  cos  a)  2  cos  (<£'  -f-  a)   ' 

and, 

_      (sin  (<f>  -J-  <£')  sin  (<£  —  i) 
N  cos  (<£'  +  a')  cos  (a  —  i) 

which  we  obtain  from  the  old  values  by 
simply  changing  a  to  ( —  a]. 

Ifc  is  to  be  observed,  for  all  cases,  when 
<£'  >  (j>  that  we  must  replace  <f>'  in  all  the 
formulas  by  <£. 

These  formulas  are  identical  with  those  of 
Bresse  ("Cours  de  Mecanique  Appliqu6e," 
Vol.  I.  3d  ed  )  and  Weyrauch,  for  the 
case  of  the  wall  leaning  forward,  the  only 
cases  examined  by  them.  Bresse  uses  the 


98 


Poncelet  method  for  the  general  case, 
which  leads  to  Poncelet's  celebrated  con- 
struction. The  routes  pursued  by  these 
authors  is  different  from  that  given  above, 
the  method  of  Weyrauch,  in  particular, 
being  much  more  complicated  5  still,  all 
three  methods  lead  to  precisely  the  same 
formula,  so  that  it  must  be  considered  as 
established  beyond  question. 

Weyrauch,  too,  in  subsequent  reductions, 
follows  Rankine  as  to  the  direction  of  the 
earth-thrust  against  the  wall,  whereas 
Bresse  takes  it  as  above.  The'case  of  the 
lt  limiting  plane  n  is  not  considered  by  either. 

43.  The  case  where  the  top  surface  slopes 
downwards  to  the  rear  is  very  rarely  met 
with  in  practice.      The  previous  formula 
apply  though  directly  on  simply  changing  i 
to  ( —  i),  since  it  is  seen  that  angle  ADB 
-  (<{>+i)  and  angle  ABD  =  90 -j-  (a—i), 

A.    AB .  i  ,    sin  (<t>-\-i] 

and  the  ratio is  now  equal  to  —          — {. 

AD  cos(tf — i) 

44.  Earth  Level  at  Top;  Back  of   Watt 
Vertical. — For  the  earth  level  at  top,  back 
of   wall   vertical,  and  <£'  =  <£   as    usually 
taken,  the  formula  (11)  takes  a  very  simple 


99 


form.      Here  we  have  <*— 0,  <f>'-—  <£,  ^=0, 
whence. 


-j 


sin  2d>  sin  d> 

* 


cos 
and 


- 


For  <£'  —  0,  which  corresponds  to  a  per- 
fectly smooth  watt,  or  otherwise  may  refer 
to  the  direction  of  the  pressure  on  a  ver- 
tical plane  in  a  mass  of  earth  of  indefinite 
extent,  level  on   top  (Art.   9),  we   have, 
when  a  —  0  and  i  =0 ,  n  =  sin  <£  and, 
17,       1  —  sin  <£      eh^ 
1  -j-  sin  <£       2 

°  —  —  ).  —  ...  (16). 

9  /        9 

The  equality  of  the  two  co-efficients  o£ 
—  in  (16)  is  easily  verified  from  the  known 

formula, 

1  —  cos  a? 
tan2  J  (x)  = 


cos  x 

by  putting  (90  —  <£)  for  x  in  both  members. 
Referring  to  Fig.  7,  and  regarding  AF 


100 

vertical,  the  top  surface  horizontal,  and 
<j>'  =  0,  we  note  that  G=~.  h2  tan  ft  and 

p 

E  —  -  h2  tan  ft  tan  (y  —  ft),  in  which  y  = 

90  —  ft.  Now,  this  result  must  agree  with 
the  right  member  of  (16),  which  is  only 

possible  when  ft  =  /  45  —  f-  \  or  2/3  = 

<!  0— ft)  5  whence  it  follows  that  for  ft'^0, 
a  —  0,  i  =  0,  as  assumed,  the  plane  of  rup- 
ture bisects  the  angle  between  the  vertical  and 
the  line  of  natural  slope. 

45.  Earth  sloping  at  the  Angle  of  Repose. — 
For  this  case  we  shall  assume  a=  0  and 
ft'  =  <£  in  addition  to  i  =  <f>,  whence  n=0 
and, 

E=^±.df.  .  .  (17), 

ft 

as  found  in  a  different  manner  in  Art.  41. 
This  simple  formula  can  likewise  be  de- 
duced directly  from  equation  (1)  of  Art. 
37,  referring  to  Fig.  7, 

E-         sin  (y  —  <ft)       __  cos  (ft  -f  ft) 
£~~  sin  (ft  -hft'  +  ^)~sin  (2  ft  + /ft)' 


101 

On  substituting  the  value  of  G,  which  is 
easily  found  for  this  case  to  be, 

_  _  1_       _etf__   sin  ft  cos  <fr  eh* 
~  cot  p—  tan  4>  T  ~cos  (ft   -f  </>)  ~2~' 

we  find  for  the  trial  thrust 

-p  _     sin  ft  cos  <£     e/ia 
"  sin  (2  <£  +  />)  ~2 

cos  <  eft2 


sin  2  <£  cot  /?  -j-  cos  2<f>    2 

Now,  by  the  reasoning  of  Art.  37,  the 
true  thrust  is  the  greatest  value  the  above 
expression  can  have,  as  fi  varies,  and  its 
greatest  value  corresponds  to  /?=90  —  <£  j 
for  then  cot  ft  is  least,  and  E  greatest,  since 
cot  ft  is  in  the  denominator.  On  substi- 
tuting this  value  a  simple  reduction  gives 
E  =  J  cos  <£  .  eh2  as  found  above  in  (17). 
Since  we  have  just  found,  for  this  case, 
that  ft  =  90  —  <j>,  it  follows  that  the  surf  ace 
of  rupture  coincides  wiih  the  natural  slope. 
The  value  of  E  from  equation  (1)  in  this 
case  assumes  the  form  0  X  oo,  since  G 
becomes  infinite  for  an  indefinitely  sloping 
surface  j  but  on  reducing  to  the  form  above 


102 

we  easily  see  the  limit  that  E  approaches 
indefinitely,  which  is  its  true  value.  The 
construction  of  Art.  39  fails  for  this  case, 
but  the  one  of  Art.  41  leads  directly  to  (17). 

46.  Pressure    of   Fluids. — The    general 
formula  (9)  above  is  true,  no  matter  how 
small  the  angle  of  repose  <f>  becomes,  and 
must  approach  indefinitely  the  expression 
for  the  pressure  of  liquids,  as  <j>  and  </>' 
tend  towards  zero  j  so  that  at  the  limit, 
for  4>  =  <£'  =  i  =  0,  we  have  the  normal 
thrust  of  a  liquid  whose  weight  per  cubic 
foot  is  e, 

E  —  |  eft  cos  a  =  1  eh2  sec  a  .  .  .  (18), 
a  well-known  formula.     By  Art.  44  we  see 
that  for  <£  =  0,  2/?  =  90,  or  the  plane  of 
rupture  approaches  an  inclination  of  45° 
as  <£  approaches  zero  indefinitely . 

47.  Mankinds  Formula  for  the  Earth-thrust 
on  a  Vertical  Plane,  in  an  Indefinite  Mass, 
sloping    uniformly.     In    Art.    9    we    have 
stated  the  conditions  that  such  a  mass 
must  satisfy  in  order  that  the  pressure  on 
a  vertical  plane,  whose  intersection  with 
the  top  slope  is  a  horizontal  line,  may  be 
parallel  to  the  line  of  greatest  declivity. 


103 

Also  in  Art.  28  we  have  seen,  that,  when 
the  wall  face  lies  below  the  limiting  plane, 
this  direction  of  the  thrust  is  the  true  one 
on  a  vertical  plane,  passing  through  the 
inner  toe  of  the  wall. 

We  have  a=  0,  <£'  =  i.  and  I  =  h,  which 
gives  in  formula  (9), 

\«~*f-T/  2  cos  i 
where, 


sin  (<f>  -f  i)  sin  (<£  — 


_  V  sin2  <f>  cos2  i  —  cos2  <f>  sin2  i 
cos  i 


_  V  cos2  i  —  cos2  <f> 

cos  i 
Whence, 

p  _  cos2  <j>  cos  i  eh2 

(cos  i  +  Vcos2i  —  cos2  <£)2'  "~2~* 

Now,  since  we  can  write, 

cos2  <£  =  (cos  i  +  Vcos2 1  —  cos2  <^>) 
X  (cos  i  —  Vcos2  i  —  cos2  <£) 
the  above  value  becomes,  on  striking  out 


104 
the  common  factor,  (cos  HVcos2  i—  cos2 


cos 

which  is  Rankine's  well-known  formula 
for  earth  pressure. 

Now  since  Rankine's  formula  was  framed 
without  the  use  of  any  assumption,  as  that 
of  a  plane  of  rupture,  and  is  accepted  as 
correct  for  the  case  in  question,  it  follows, 
that,  when  the  pressure  is  assumed  to  be 
parallel  to  the  surface,  the  assumption  that 
the  surface  of  rupture  is  a  plane  will  give 
correct  results,  and  can  be  safely  used  in 
the  graphical  method  which  is  absolutely 
dependent  on  it. 

It  will  be  observed  that  formulae  (16) 
and  (17)  can  be  deduced  directly  from  (19) 
by  making  i  =  0  and  i  =  <£  respectively. 
Rankine  has  given  a  simple  graphical  con- 
struction of  the  last  fraction  in  (19)  in  his 
"  Civil  Engineering/7  which  saves  labor  in 
computing. 

48.  Unit  Pressures  on  a  Vertical  Plane 
at  Depth  x  below  a  uniformly  Sloping  Sur- 
face, the  Direction  of  the  Pressure  being 


105 


taken  Parallel  to  the  Line  of  Greatest  De- 
clivity.—As  in  Art.  35  we  shall  consider  a 
wedge  of  thrust  of  infinitesimal  dimen- 
sions, of  which  the  left  face  A  B  (Fig.  10) 
is  vertical,  and  the  upper  surface  paral- 
lel to  the  top  slope.  The  weight  of 
the  vertical  prism  that  rests  upon  any 
trial  base  as  BC  is,  e  .  BC  .  cos  i  .  x  .  = 
AT  .  BC .  ex  I A  B  (Fig.  8);  so  that  neglecting 
the  weight  of  the  infinitely  small  wedge 
ABC  we  get  the  value  of  E  from  equa- 
tion (1)  of  Art.  37  by  simply  replacing  G 
by  this  value.  Equation  (2)  of  Art.  38  is 
thus  replaced  by 

F-^L    AT     BCCI 
-AB.AT.  BCTr 

which  is  exactly  that  given  in  Art.  38  mul- 
tiplied by  the  constant  2x1  AB.  All  the 
subsequent  reductions,  therefore,  hold  if  we 
simply  put  h=AB  in  the  final  equations, 
and  multiply  the  result  by  2x/AB.  Hence 
divide  (19)  by  AB  =  h  and  change  the 
coefficient  eh/ 2  to  ex,  to  find  the  intensity 
of  the  pressure,  E  +  AB,  at  a  depth  x; 
and  on  integrating  this  expression,  mul- 
tiplied by  dx,  between  the  limits  o  and  h, 


106 


we  are  at  once  conducted  to  (19),  which 
is  thus  .proved  true  by  the  method  of 
integration  of  the  effects  of  earth  particles, 
which  is  independent  of  the  assumption  of 
a  plane  surface  of  "upture  ^xcending  to 
the  surface. 

Precisely  the  scum,  conclusions  hold  for  a 
vertical  tvall,  or  one  leaning  forwards,  ivlien  E 
is  assumed  to  maJce  the  angle  <£'  or  <f>  with  the 
normal  to  the  wall,  since  G  is  simply  replaced 
as  before  by  the  weight  of  the  vertical 
prism  for  a  uniform  top  slope,  and  ultimately 
we  replace  h2  by  2x  in  the  general  formula 
(11)  to  get  the  intensity  of  pressure  in  the 
direction  given,  at  the  depth  x  from  the 
surface,  so  that  on  integrating  as  before  we 
deduce  (11)  without  the  necessity  of  con- 
sidering the  surface  of  rupture  as  extend- 
ing to  the  surface.  The  graphical  method, 
using  this  hypothesis,  should  again  give 
good  results.  It  is  possible  though,  in  this 
case,  that  the  influence  of  the  wall  friction 
may  have  some  effect  in  deflecting  the 
weights  of  the  vertical  prisms  from  a  ver- 
tical line  ;  for,  when  it  is  so  transmitted,  the 
usual  direction  of  the  pressure  is  parallel 


107 


to  the  surface  (Art.  9).  For  walls  leaning 
backwards  the  prisms  do  not  rest  vertically 
over  the  bases  of  the  prisms  of  thrust,  and 
the  theory  would  seem  to  be  inapplicable  ; 
so  that  the  formulae  for  this  case,  (8)  and 
(9),  have  to  rest  upon  the  unproved  hypo- 
thesis of  a  plane  surface  of  rupture  extend- 
ing  to  the  surface,  and  may  depart  consid- 
erably from  the  truth.  We  conclude,  that, 
except  for  the  cases  for  which  -Rankine's 
formula  is  applicable,  the  plane  surface  of 
rupture  is  still  an  unproved  hypothesis. 

49.  Pointof  Application  of  the  Thrust;  Uni- 
form Slope. — We  have  the  normal  compo- 
nent of  the  thrust  on  a  wall,  by  (9)  whether 
the  wall  inclines  forward  or  backward  or 
is  vertical,  expressed  by  the  relation, 

E1  =  (9)  X  cos  #  =  cP, 
c  being  constant  •  whence  the  thrust  on 
the  area  dl  X  1  is  nearly 

dEi  =  2cm, 

and  the  distance  from  where  the  inner 
face  of  the  wall  interesects  the  top  slope 
to  the  centre  of  pressure  is  equal  to 
the  limit  of  the  sum  of  the  elementary 


108 


pressures  multiplied  by  tlieir  arms,  divided 
by  the  total  pressure,  or, 


f 

*J     I 


hence  the  centre  of  pressure  on  the  wall  is 
1/4  vertically  above  the  base. 

50.  Surcharge  uniformly  distributed. — If 
the  filling  of  height  li  has  a  horizontal 
surface  upon  which  a  uniform  load  of  any 
kind  rests,  replace  its  weight  by  that  of 
an  equivalent  quantity  of  earth,  giving  the 
total  load  the  same,  and  call  the  height  of 
the  reduced  load  h'.  The  total  pressure  on 
the  vertical  wall  of  height  his  now  by  (11), 
E  =  Ke  ((h  +  h')2  —  h1-}  =  Keh  (h  -f  2k1), 
whence, 

dE  =  Ke.2  (h  -f  h')  dh  ; 
and  the  distance  of  the  centre  of  pressure 
from  the  top  of  the  wall  downwards  is, 


2  f 

•J  o 


'*')  hdh 


h  (h  +  2h')          3\ 
or  from  the  base  of  the  wall  upwards, 


109 


Ji- 


h2 -f 
37*  +  07* 


1  + 


ll  +  27/7  a 


It  is  more  than  probable  that  the  theory 
for  this  case  will  prove  illusory  in  practice, 
and  will  give  a  large  excess  of  pressure ; 
so  that,  most  frequently,  such  surcharged 
loads  are  ordinarily  allowed  for  by  a  large 
factor  of  safety,  particularly  where  the 
earth  is  bound  by  cross-ties,  stringers,  etc., 
or  the  surcharge  is  not  free  to  move  later- 
ally as  well  as  vertically. 

Surcharge 


.   !O(a). 


In  the  case  of  sea  walls,  the  backing  is 
saturated  with  water  at  high  tide,  up  to  a 
certain  level  BF,  fig.  10  (a),  so  that  it  is 
well  to  ignore  the  friction  at  the  back  of 


110 

the  wall  on  BC  and  for  additional  safety 
it  will  be  neglected  on  the  portion  AB. 

Call  the  weight  of  the  backing  per  cubic 
foot  above  BF,  e1  and  the  angle  of  repose 
<#>!.  The  corresponding  quantities  for  the 
saturated  backing  below  J?jF  will  be  desig- 
nated by  62  and  <£o.  The  value  of  <£o 
should  be  found  by  experiment  and  €2 
computed  as  explained  at  p.  31.  If  the 
water,  at  high  tide,  is  at  the  same  height 
on  the  front  and  back  faces  of  the  wall, 
the  water  pressures  on  those  faces  will 
balance  and  need  not  be  considered. 

Let  A J5— #! and E^ Dearth  thrust 
—  Ji2  and  £2= earth  thrust  on  BC. 


By  Art.  44  (16),  K  —  -J.  Tan2   45°-_ 

therefore  if  ~h'e^—  W=  surcharge  or  load  in 
pounds  per  square  foot,  on  AD ;  by  the 
analysis  just  given, 

EI  =  Kejii  (hi  +  2ft1)  =  tan2  /45°  —  ^\ 

\^  +  *iWl  and  ^  acts  above  B,  a 
distance, 


Ill 


Next,  assuming  that  the  load  on  the  hori- 
zontal plane  BF  is  uniform  and  Wo  Ibs. 
pr.  sq.  ft.  and  calling  the  height  of  this 
load  reduced  to  the  specific  gravity  of  the 
earth  below  BF,  7?o, 


Hence  as  before, 
E,  =  tan*  (45°  -  *»). 
and  EI  acts  above  (7,  a  distance, 

d  +    ^  V*2 

\     r  /^2+2  V  3 

If  no  surcharge  is  considered,  the  formu- 
las apply  on  making, 

TF=o  h'=o  .  *  .  TPo= 


In  either  case  having  found  JJX  and  E^  in 
magnitude  and  position,  the  position  of 
the  resultant  E\  -f-  E^  can  be  found  by 
taking  moments  about  C. 

The  above  formulas  will  be  found  to 
reduce  to  those  given  by  Mr.  D.  C.  Serber, 
in  Engineering  News,  Aug.  23,  1906.  It 


112 

is  stated  there,  that  the  Department  of 
Docks  of  New  York  City  specify  a  sur- 
charge of  1,000  Ibs.  pr.  sq.  ft.,  acting  as  a 
vertical  load. 

51.  Moments  of  the  Thrust  about  the  Inner 
Toe  of  the  WaW.— Let  us  decompose  the 
thrust  E  against  the  wall  into  two  compo- 
nents, E\  and  Ez,  respectively  normal  to 
and  acting  along  the  inner  face  of  the  wall. 
If  E  makes  the  angle  <f>'  with  the  normal 
to  the  wall,  we  have,  from  E  =  Keh2, 

Ei  =  E  cos  <£'  =  K  cos  <}>'.eh2- 

or  putting,  K\  =  K  cos  <j>' 

we  have,      EI  =  K\  eh2  • 

also,  E%  —  E  sin  <f>'  =  EI  tan  <£'. 

It  is  understood  in  these  formulae,  that, 
when  <f>'  >  <£,  we  must  replace  <f>'  by  <f>. 

If  the  inner  face  of  the  wall  makes  an 
angle  a  with  the  vertical,  we  have  the 
thrust  acting  at  a  distance  cl  =  ch  sec  a 
from  the  inner  toe  of  the  wall,  where  c  = 
^  by  theory  for  a  uniform  slope  ;  there- 
fore, the  moment  M  of  the  thrust  about  the 
inner  toe  of  the  wall  is  EI  cl}  since  the  mo- 


113 

ment  of  E^  is  zero  j  or  putting  for  abbre- 
viation, 

m  =  c  K\  sec  a 
we  have, 
M  —  E\  ch  sec  a  =  c  K\  sec  a  .  eh*  —  meh*. 

In  subsequent  investigations  it  is  well  to 
recall  that  h  represents  the  vertical  height 
from  the  inner  toe  of  the  wall  to  where  the 
line  of  the  inner  face  pierces  the  top  surface 
of  the  earth  backing,  and  that  e  represents 
the  weight  per  cubic  foot  of  earth. 


CHAPTER   IV 

EXPERIMENTAL  METHODS.  COMPARISON  WITH 

THEORY.       THE    PRACTICAL    DESIGNING 

OF    RETAINING     WALLS 

52.  Many    experiments   have    been   re- 
corded pertaining  both   to   retaining-walls 
proper  and    to    rotating    retaining-boards. 
Where  the  backing  is  of  dry  sand,  possess- 
ing little  or  no   cohesion,   the  results,  for 
the    retaining-walls    proper,    agree    fairly 
well    with    the    theory    advanced    in    this 
book,  which  includes  all  the  wall  friction 
that  can  be  exerted,  especially  where  the 
walls  were  several  feet  in  height;   but  they 
do  not  agree  with  the  Rankine  theory,  in 
which  the  direction  of  the  pressure  on  a 
vertical  plane  is  always  assumed  parallel 
to  the  earth  surface. 

53.  The  results  for  some  of  the  exper- 
iments on  model  walls  at  the  limit  of  stabil- 
ity are  given  in  the  adjoining  table. 


115 


;•••] 

1  1  M 

£  g-33  0-i.g  | 
|5.S8lH! 

f     S    1    ^      "      0)     ,,     § 

ex 

1     i 

s  "  «  1  ^^I-S 

*o   w  ^       js       "^ 
5"            D  a  S  S  3 

* 

O             O             0 

i-lMi^l 

=S|^5|S- 

-0- 

0     O     O      0     O     O     O 

O5  CO  CO  ^O  *O  iO  CO 
CO  CO  CO  CO  TJH  CO  CO 

r«     *0        3^^        °     •§    ^3 

™    QJ     co     >     fe  ^     »H 
S  •"!                        o3     53     fl 

fllllSll 

•- 

OOOOiOOO 

v's^l^as 

i»"  -e-      o  be  «g  §  w> 

ItfllllAi 

8 

CO 
00000£,S} 

t  I 

.s  s^  «-^^s 
^^-•5fl^?- 

liis^l^l 

^  o       B  5  2  **  ^ 

,, 

O  iO  X  O5  (M  C^l  (M 

(M  05  T-H  (M  0  CD  CD 

|  S*8  §1-1  3  | 

(NOCO  COOOO 

X^'o'cJ     S     0^     M)0 

•»*  "S 

§!§§§§? 

^  «|  ^^-g.s^ 
«ll«s  all- 

*• 

1-1  i-i  O  O  O  i-t  »H 

s  §  -c  °  .s  ii  §  t» 

*  1 

-*'  0  0  0  iM  !N  rH 

IIJ11  "83 
ifljflfi 



I'    £.  ^5  ^    «   t>    °    S 

Authority. 

|    .  6    .    .    .    . 

riltiti 

W.-1HOOOO 

z-3  P>  B'*M 

a  *s  \*-  §.  9  ••  «  • 

"S^i^ajoStn 

iifiiiifi 

irJPl^ 

6 
fc 

r-t  C^  CO  •*  ^>  CD  l> 

^^oj'oj0     ^o;r3 

•slilllP^ 

116 

The  walls  were  all  vertical  walls  of  rectangular 
cross-section,  except  the  last  two,  which  were 
peculiar  wooden  triangular  frames  whose  inner 
faces  made  angles  27°  30'  and^55°  respectively 
with  th<;  vertical. 

In  No.  6,  the  face  coincided  with  the  "  limiting 
plane  "  (Arts.  28  and  41)  and  in  No.  7  was  below 
it.  in  either  case,  the  thrust  was  first  found  on  the 
vertical  plane  through  the  foot  of  the  inner  face 
and  this  was  combined  with  the  weight  of  the 
earth  over  the  face  and  the  weight  of  the  frame  to 
find  the  resultant  on  the  base  (see  Art.  32). 

Wall  No.  1  was  of  pitch-pine  blocks,  backed  by 
macadam  screenings,  the  level  surface  of  which 
was  3  inches  below  the  top  of  the  wall.  Wall 
No.  5,  of  brick  in  Portland  cement,  was  a  sur- 
charged one;  the  level  upper  surface  of  the  sur- 
charge being  4.26  ft.  above  the  top  of  the  wall,  the 
surcharge  extending  entirely  over  the  top  of  the 
wall  at  45°  to  the  horizontal.  In  the  other  walls, 
the  earth  surface  was  level  with  the  top  of  the  wall. 
Wall  No.  2  was  of  bricks  laid  in  wet  sand;  No.  3, 
of  wood,  and  No.  4  was  of  wood  coated  on  the 
back  with  sand. 

54.  Elaborate  experiments  on  rotating 
retaining-boards,  backed  by  sand,  have 
been  made  by  Leygue  ("  Annales  des 
Fonts  et  Chausse'es,"  Nov.,  1885),  Darwin 
and  others,  which,  in  the  earlier  editions 
of  this  work,  were  given  in  detail.  They 
are  omitted  here,  since  they  have  been 


117 

discussed  by  the  writer  very  fully  in  an 
article  entitled  "  Experiments  on  Retain- 
ing-walls  and  Pressures  on  Tunnels."* 
The  conclusion  was  drawn  that  the  results 
can  be  harmonized  with  theory  by  includ- 
ing the  influence  of  cohesion.  The  dis- 
cussion involved  a  complete  theory,  mainly 
graphical,  of  earth  pressure,  where  the 
earth  is  supposed  endowed  with  both 
friction  and  cohesion. 

As  regards  the  experiments  of  Leygue 
on  rotating-boards,  it  was  found  that, 
assuming  an  adhesion  or  cohesion,  of  only 
about  1  Ib.  per  sq.ft.,  for  the  dry  sand 
used,  the  experiments  were  in  harmony 
with  theory;  but  that  the  results  differ 
essentially  from  the  usual  theory  where 
cohesion  is  neglected.  The  discrepancies 
were  proved  to  be  due  entirely  to  the 
small  size  of  the  models  used  and  it  is 
suggested  that  in  future,  walls  of  6  feet 
and  upwards  in  height  be  experimented 
on,  where  the  influence  of  a  cohesion  of 
only  1  Ib.  per  sq.ft.  is  very  small  and  can 
be  neglected  in  the  analysis. 

*  Transactions  Am.  Soc.  C.E.,  Vol.  LXXII, 
p.  403  (1911). 


118 

55.  Center  of  Pressure.  Leygue,  in  the 
experiments  on  retaining-boards,  found  the 
moment  of  the  earth  thrust  about  the  toe 
and  also  determined  the  surface  of  rupture. 
Using  the  corresponding  wedge  of  rupture, 
the  writer  computed  the  thrust  and  its 
normal  component.  On  dividing  the  mo- 
ment given  by  the  latter,  the  quotient 
gives  the  distance  of  the  center  of  pressure 
of  the  earth  thrust  from  the  base.  It 
was  found  to  lie,  as  an  average  for  all  the 
experiments,  at  0.34  height  of  the  board  in 
contact  with  the  filling  for  dry  sand  and 
0.405  height  for  millet  seed.  For  sand, 
the  values  varied,  for  a  vertical  wall  from 
0.319  per  earth  surface  horizontal  to  0.346 
for  the  surface  sloping  at  the  angle  of 
repose.  For  boards  leaning  towards  the 
earth,  when  tan  a  (Fig.  10,  p.  94)  was 
+  i  the  variation  was  from  0.296  to  0.337; 
for  tan  «  =  +§  from  0.325  to  0.375.  For 
boards  leaning  from  the  earth,  tan  a  =  —  f , 
the  variation  was  from  0.352  to  0.363. 

These  results  are  approximate,  for  al- 
though the  exact  prisms  of  rupture  were 
used,  the  chord  of  necessity  replaced  the 
true  curved  line  of  rupture  in  the  construe- 


119 


tion  of  Fig.  3,  p.  35,  and  cohesion  was  neg- 
lected. The  effect  of  cohesion  is  to  lower 
the  center  of  pressure;  so  that  doubtless 
for  sand  absolutely  devoid  of  cohesion,  the 
atios  should  be  larger.  However,  from 
lack  of  more  complete  observations  on 
large  models,  the  theoretical  value  £  will 
be  used  in  the  computations  below. 

56.  The  center  of  pressure  for  a  sur- 
charged wall  of  the  type  shown  by  Fig.  4, 
p.  40,  only  with  the  back  vertical  and  the 
each  surface  extending  from  C,  the  top 
of  the  inner  face,  at  the  angle  of  repose, 
<£=33°41',  to  the  level  surface  2-6,  has 
been  found  by  the  writer  *  for  various 
ratios  of  ti  to  /?,  where  h=  height  of  wall, 
h'  =  height  of  surcharge  above  the  top  of 
wall  and  c=  vertical  distance  from  foot 


h' 

c 

h' 

e 

h 

h 

00 

0.333 

1.00 

0.364 

2.00 

0  .  353 

0.75 

0.364 

1.50 

0.356 

0.50 

0.364 

1.25 

0.360 

1.11 

0.362 

0.00 

0.333 

*  Trans.  Am.  Soc.  C.E.,  Vol.  LXXII,  p.  410. 


120 

of  wall  to  the  center  of  pressure,  divided 
by  the  height  of  the  wall. 

In  finding  the  values  of  c,  h'  was  taken 
as  10  feet  and  the  earth  thrusts  on  walls  of 
heights,  1,  2,  3,  .  .  .,20  ft.,  were  found  by 
the  construction  of  Fig.  8  (see  pp.  88-89). 
By  subtraction,  the  earth  thrust  on  each 
foot  of  wall  was  obtained,  and  by  taking 
moments  about  convenient  points,  the 
centers  of  pressure  for  heights  of  wall 
varying  from  5  to  20  feet  were  easily 
obtained  and  c  computed,  as  given  in  the 
table. 

57.  From  a  discussion  of  all  the  exper- 
iments, the  conclusion  was  drawn  that  the 
sliding-wedge  theory,  involving  wall  fric- 
tion, is  a  practical  one  for  the  design  of 
walls  backed  by  granular  materials  and 
subjected  to  a  static  load.  Often,  however, 
in  practical  design,  vibration  due  to  a 
moving  load  has  to  be  allowed  for;  also 
the  effect  of  heavy  rains.  Both  these 
influences  tend  generally  to  lower  the 
coefficient  of  friction  and  add  to  the  weight 
of  the  filling.  To  allow  for  these  in- 
fluences, in  designing,  the  normal  com- 
ponent Ei  of  the  earth  thrust  will  alone  be 


121 

multiplied  by  a  factor  of  safety  (7,  the  fric- 
tion Ei  tan  <f)',  exerted  downwards  along  the 
back  of  the  wall,  remaining  unchanged. 
This  allows  very  materially  for  a  decrease 
in  </>'  due  to  rains  and  vibration,  as  well  as 
for  an  increase  in  the  thrust.  A  factor  of 
safety  (7=3.5  is  suggested  for  walls  6  ft. 
high,  decreasing  to  3  for  walls  10  ft.  and  up- 
wards. For  walls  50  ft.  high  and  upwards, 
or  for  lower  walls  with  a  high  surcharge,  this 
factor  may  possibly  be  still  further  de- 
creased, since  before  the  embankment  is 
finished,  the  cohesive  and  chemical  actions 
in  the  earth  have  doubtless  consolidated  it  to 
such  an  extent  that  the  actual  thrust  is 
much  less  than  the  computed  one  when 
cohesion  is  neglected. 

In  any  case,  the  true  thrust  E  (not 
multiplied  by  any  factor)  when  combined 
with  the  weight  of  the  wall,  must  give 
a  resultant  that  will  pierce  the  base  within 
its  middle  third,  since  it  is  desirable  that 
pressure  should  be  exerted  over  the  whole 
base.  If  this  does  not  obtain  for  a  certain 
type  of  wall,  the  base  should  be  made 
wider. 

If  heavy  loads,  as  railway  trains,   pass 


122 

over  the  surface  of  the  filling,  near  a  re- 
taining-wall,  the  weight  of  the  load  should 
be  replaced  by  an  equal  weight  of  earth 
and  the  earth  thrust  determined  as  in 
Art.  50  or  by  aid  of  the  construction  of 
Fig.  8,  p.  83,  or  that  of  Fig.  4,  p.  40. 

With  an  earth  foundation,  a  footing  of 
masonry,  projecting  beyond  the  wall,  should 
be  built  of  such  width  that  the  true  re- 
sultant on  the  base  should  pass  near  its 
center.  This  should  totally  prevent  the 
increased  leaning  with  time  sometimes 
observed.  Lastly,  to  ensure  against  slid- 
ing, the  base  should  be  inclined. 

58.  General  Formula  for  Stability  of 
Retaining-walls  against  Overturning.  Let 
Fig.  11  represent  a  wall  A  BCD,  whose 
length  perpendicular  to  the  plane  of  the 
paper  is  unity  and  whose  exterior  and 
interior  faces  and  diagonal  AC,  make 
angles  with  the  vertical  equal  to  ft,  a  and 
w  respectively.  Let  W  denote  the  weight 
of  the  wall  and  g  the  horizontal  distance 
from  its  line  of  action  to  the  outer  toe  A ; 
also  call  <7  the  factor  by  which  it  is  necessary 
to  multiply  the  normal  thrust  Kieh2, 
leaving  the  friction  fKieh*  at  the  back 


123 


of  the  wall  constant,  in  order  that  the 
resultant  on  the  base  may  pass  through 
the  outer  toe.  Here  /=tan  </>'  (when 


Fig.  11. 

<£'>  4>,  replace  <£'  by  </>)  and  the  quantities 
h,  t,  e,  w,  i,  <f>  and  </>'  have  the  meanings 
given  in  Art.  52. 

Taking    moments   around   A,  we    have, 

Wg+fK1eh*tcosa  = 

<rKieh*(ch  sec  a+t  sin  «). 

We  find  also,  t  =A(tan  co  -tan  a) ;   and  since 


124 

the  moment  Wg  is  equal  to  the  sum  of 
the  moments  of  the  triangular  prism  ADI 
and  the  rectangular  prism  IDCE,  minus 
the  moment  of  the  triangular  prism  BCE, 
all  of  the  same  density  w,  we  readily 
find  it  to  equal, 

2  h* 

—  tan  /3'—  h  tan  0-f—  (tan2  w-tan2  /3) 
-  o  - 

/i2  1  1 

— —  tan  a.   h(tan    w-—  tan  a)  \w; 

Z  6  J 

or, 

wA3 

-—(3  tan2w  —3  tan  w  tan  «-|-tan2a  —  tan20). 
6 

On  substituting  the  values  for  t  and  Wg 
and  resolving  with  respect  to  tan  o>,  we 
find, 

tan2  w-f 

r «  i 

tan  w   2—Ki(f  cos  a  -<r  sin  a)  —tan  «    = 

e 

—2Ki  [<TC  sec  a+tan  a(f  cos  a— <r  sin  <r)J 
ic 

--(tan2  «-tan2/3). 
o 


125 


This  formula  equally  applies  when  the 
inner  face  of  the  wall  leans  away  from  the 
earth,  or  B  falls  to  the  right  of  E, 
on  simply  replacing  sin  a  and  tan  a  by 
(—sin  a)  and  (—tan  «)  throughout.  As 
this  formula  is  independent  of  h,  it  is  true 
for  all  values  of  h.  When  h  is  given,  tan 
to  is  found  from  the  formula,  whence, 
t  =h  (tan  w  —tana). 

59.  Since  J/A=(tan  co-tan  a),  if  we 
take  h=l,  the  value  of  t=AB  correspond- 
ing, represents  the  ralio  of  the  thickness 
of  the  base  to  h  for  any  height  of  wall. 
Hence,  for  simplicity  in  the  following 
applications  to  the  various  types  1,  2,  3, 
4,  5,  Fig.  12,  the  thicknesses  at  top  and 
bottom  and  the  volume  will  be  computed 


for  h  =  l.     The  natural  slope  will  be  taken 
at  3  base  to  2  rise  or  0=33°  41'  and  it 


126 

will  be  assumed  that  </>'=</>,  (whence 
/=tan  <£'=f),  c=£  and  <r=3,  which  refers 
to  walls  10  feet  high  and  upwards. 

The  tables  given  below  are  computed 
for  two  ratios  of  specific  weight  of  earth 
to  wall:  e/w  =|  and  e/w  =  £,  corresponding, 
perhaps,  to  concrete  and  good  brick  walls 
respectively. 

60.     Type  I.     Vertical  Rectangular  Wall. 
\ 

a=0,  0=0,  J=tan  co. 
The  general  formula  of  Art.  58,  reduces  to 

P+^KJt  =-2#!. 

w  w 

When  i=0,  from  p.  99,  we  have,  since 
Ki=K  cos  </>'  =K  cos  0* 


*  The  computation  of  K,  for  some  of  the  types, 
by  formulas,  being  very  long,  the  graphical  method 
of  Art.  39  can  be  substituted  for  it.  Thus  in  Fig. 
8,  let  e  =1  and  lay  off  h  =  vertical  projection  of 
AF  =1  foot  (say  to  a  scale  10  inches  to  1  foot) 
and  draw  from  B,  now  coinciding  with  F ,  a  hori- 
zontal line  to  represent  the  earth  surface;  then 
exactly  as  indicated  on  p.  88,  locate  the  points 
O,  I,  C,  H.  The  thrust  E  =  Keh*  =K  =  \CI.CH.:  Ki 
**K  cos  </>.  When  i=tf>,  AD±  GO  ,  AI  ~  oo  and 


127 
cos2  <b 


0.109. 


!=- 

2  (l+V2sin0) 

.'.  for   e/w=l,  t  =0.334, 
e/u?=$,  t  =  0.363. 

When  i  =  0,  page  100,  A':  =  £  cos2  (f>  =0.346. 
Whence,  for  e/w=%,  t  =0.541;  e/w  =£,  t  = 
0.583. 

61.   Type  2.     Vertical  Back.     Front  face 
battered  at  2  inches  to  the  foot. 
«=0,  tan  0  =  j-,    0=9°   28'.     The  formula 
reduces  to, 

-2Ki+-t&n*  0. 
10  3 


AC=AD;  hence  the  point  7  can  be  taken  any- 
where on  AD.  With  i  =0  and  C  located  as  before 
.-.  as  above,  Ki  =  \CI.CH  cos  0. 

In  type  5,  the  earth  pressure  on  the  wall  was 
taken  as  making  the  angle  0  with  its  normal. 
The  assumption  was  only  intended  for  usual 
batters  of  leaning  walls,  say  a<10°,  for  which 
it  is  practically  correct.  For  large  values  of  a, 
the  assumption  is  not  to  be  made,  the  error  increas- 
ing with  the  angle  a. 


128 

For  i  =  0,  as  above,  Ki  =0.109. 
.'.  e/w=\,  *2+0.097Z  =0.144+0.0093, 
/.  t  =0.346. 

e/w  =i  *2+0.116*  =0.174+0.0093, 
.'.   t=  0.374. 

For  t=0,  /d  =0.346,  Art.  60. 
/.  e/w=l  t  =0.548;   e/w=$,  t  =0.589. 


62.  T?/pe  3.  5o^  /aces  battered  2  inches 
to  the  Foot.  On  replacing  sin  a  by  (  —sin  a), 
tan  a  by  (-tana)  in  the  general  formula, 
and  noting  that  here,  tan2  «=tan2  /3, 


tan2  w+-2Jft:1(/  cos  a  +  o-  sin  «)+tan  a\ 

e     r 

tan  o>  =—2/^i   a|  sec  «—  tan  a  (/cos  « 

+  0-  sin  a)    . 


Formulas  (13)    and    (14)    p.    97,  give,   for 

0  =  0'  =33°  41',  i  =0,  a  =9°  28';  n  =0.8434, 

#,=0.143. 

For    e/w=l,  tan2  w+0.387,  tan  co  =0.157. 

.-.  tan  co  =0.247.-.  t  =tan  co+tan  a  =0.414. 


129 

For    e/w-it  t  an  2co-f  0.430,  tan   co  =0.189. 
/.  tan  w  =0.270.'.  t=  0.437. 

63.  Type  3  continued.  Let  i  =  0.  In 
this  case,  the  "  limiting  plane "  of  Art. 
28  concides  with  the  vertical  AO  of  Fig. 
5,  p.  50.  Since  the  inner  face  of  the  wall 
AB,  Fig.  5,  lies  below  it,  the  thrust  on 
AO  =  T=%e  cos  </>,  AO2  (acting  parallel  to 
BO)  must  now  be  combined  with  the 
weight  of  the  earth  A  BO  to  find  the  re- 
sultant on  AB.  Taking,  as  before,  the 
vertical  height  h  of  AB  =1,  we  find 
AO  =1.111  and  for  e=l,  !T  =0.513.  On 
combining  graphically,  this  thrust  on  AO, 
making  the  angle  <£  with  the  horizontal 
with  the  weight  of  ABO(e  =  1),  we  find  the 
resultant  thrust  on  AB  =  0.570  and  that 
it  makes  an  angle  32°  03'  with  the  normal 
to  A  B.  We  have  to  substitute  in  the 
general  formula  /  =  tan  32°  03' =0.626; 
also  the  normal  component  of  the  thrust 
=  0.570  X  cos  32°  03' =0.483.  As  this 
corresponds  to  the  assumed  height  of 
AB=h=l  and  e  =  l,  it  is  the  value  of  K\. 
Whence  substituting  Ki  =0.483,  /  =0.626* 

*  Formulas  have  been  derived  by  the  writer  for 


130 

in  the  formula  of  Art.  62,  we  have, 

for   e/w=l,  tan2  co+0.881    tan  co  =0.535. 

.'.  tan  «  =0.414,  t  =0.580. 
Fore/w=-£,  tan2  «  +  1.0024  tan  co  =0.642. 

.-.   tan  co  =0.439.     /.  t  =0.606. 

64.  Type  4.  Front  Face  Vertical,  Inner 
Face  Battered  2  Inches  to  the  Foot.  The 
moment  formula  differs  from  that  of  Art. 
62  only  in  the  addition  of  the  term 
(—  |  tan2  a)  to  the  right  member.  Hence, 

Ki  and  /  for  any  value  of  i,  but  the  worlj  is  too  long 
to  be  given  here.  The  results  for  t  =</>  will  be 
stated. 

From  the  formula, 


[-(«•  -i)] 


*•>- 

-tan    450  +      -a    tan* 

compute  e.  In  this  instance,  e  =48°  34'.  The 
thrust  on  the  wall  AB  makes  with  the  normal 
to  the  wall,  the  angle, 

7  =90°  -(e  +«)  =90°  -58°  02'  =31°  58'; 
whence          /  -tan  31°  58'  =0.624. 
The   value   of   K\   is  now   given   by   the   formula, 
_cos  7  cos  (0  —a)  tan  a 
2    cos  (<f>  +e)  cos  a' 

which  for  7  =31°  58';  e  =48°  34',  <j>  =33°  41', 
a  =9°28',  reduces  to  Kt  =0.483,  as  found  graphically. 


131 

we  at  once  derive,  when  i=0:Kl  =0.143 
(Art.  62),  for  e/w  =  f,  tan2  co+0.387  taa  «  = 
0.148. 

/.  tan  co  =0.237  /.  £=tan  co+  tan  a  =0.403; 
for  e/w  =£,  tan2  co+0.430  tan  co  =0.180. 
.-.  tan  co  =0.260  /.  £=0.260+0.167=0.427. 
When  i=<f>,  as  in  Art.  63,  K^  =0.483, 
/=  0.626,  and  the  moment  formula  just 
quoted  reduces  to: 

e/w=l,  tan2  co+0.881  tan  co  =0.526;  v 
/.  tan  co  =0.408,  t  =tan  co+tan  a.  =0.574/ 
e/w  =  f,tan2  co  +  1.024  tan  co  =0.633. 
.-.   tan  co  =0.434,  #=0.601. 

65.  Type  5.  Leaning  Wall.  Front  Face 
Battered  2  Inches  to  the  Foot,  Rear  Face 
Parallel  to  the  Front  Face.  The  formulas 
of  p.  96  are  now  applicable  for  computing 
K!  =K  cos  0.  For  <£=</>'  =33°  41',  i=0, 
«  =9°  28',  we  derive  n  =0.7544,  K,  =0.081. 
Putting  a  =/3  =9°  28',  the  moment  formula 
is 


, 

tan2 


\e  1 

—  2Ki(/cos  a—  <rsin  a)  -tana   tan  co 

e     W[<r  1 

—  21^   —  sec  a-ftan  a  (/  cos  a  —or  sin  a)    . 
w        Lo 


132 

Taking  as  before,  a  =3,  /  =  f,  a  =  9°  28'  =/3, 
when  e/w=l,  tan2  co  -0.149  tan  co  =0.112. 
/.  tan  co  =0.416.  .-.  £=tan  <a  -tan  <*  =0.249. 
For  e/w=%,  tan2  co  -0.144  tan  co  =0.135. 
/.  tan  co  =0.446.  /.  £  =0.446  -0.167  =0.279. 
Assuming  i  =  <£,  the  formulas  of  p.  96,  give, 


2   COS2aCOS(<p  —a) 

whence  HTi=  0.250. 

/.  e/w=f,  tan2  w  -0.112    tan    w  =0.347; 
/.  tan  co  =0.648.     .-.  «=  0.481; 

e/iy  =|,  tan2  co  -0.101  tan  co  =0.416. 
/.   tan  co  =0.697,  ^  =  0.530. 

66.  As  a  check  on  the  computations, 
the  values  of  Ki,  for  all  the  cases  discussed, 
were  likewise  found  by  the  graphical  con- 
struction of  Fig.  8,  p.  83.  Then,  Fig.  11, 
the  resultant  of  the  components  SKieh2 
and  fKieh2,  for  h  =  l,  was  combined  with 
the  weight  W  of  the  wall,  acting  through 
its  center  of  gravity,  to  find  the  resultant 
on  the  base.  In  every  instance,  it  passed 
nearly  or  exactly  through  the  outer  toe. 

The  next  step  was,  assuming  o-  =  1  , 
to  combine  Kieh*  and  fKieh2  (h  =  l),  Fig. 
11,  to  find  the  true  resultant  on  J5C, 
which  was  then  combined  with  W  to  find 


133 

the  true  center  of  pressure  on  the  base 
A B  of  the  wall.  Call  a  the  distance  from 
this  center  of  pressure  to  the  center  of 
the  base  AB;  then  the  ratio  a/t,  was 
computed  and  inserted  in  the  following 
table,  which  contains  the  results  of  the 
above  computations.  When  a/Z<0.167, 
the  true  center  of  pressure  on  the  base  is 
within  the  middle  third  limit,  so  that  the 
whole  base  is  in  compression;  when  a/t  = 
0.167  there  is  no  stress  at  the  inner  toe, 
and  when  a/t>  0.167,;part  of  the  base  only 
is  in  bearing.  The  ratio  a/t  will  be  counted 
positive  or  negative  according  as  the 
resultant  on  the  base  meets  it  to  the  left 
or  to  the  right  of  its  center. 

It  will  be  observed,  for  cases  one  and 
three  of  type  4,  that  a/t>  0.167.  In  the 
first  case,  increase  t  from  0.403  to  0.417; 
in  the  second  case,  from  0.427  to  0.432. 
These  values  are  inserted  in  the  table  in 
parentheses.  The  resultant  on  the  base, 
in  each  case,  will  then  cut  the  base  \t 
from  the  outer  toe.* 

'  *  If  the  resultant  on  the  base  of  the  wall  for 
the  actual  thrust  (v  =1)  is  to  pass  \  base  from 
the  outer  toe,  then  for  the  leaning  wall  shown 


134 

67.  In  the  last  column  of  the  table, 
is  given  the  angle  that  the  true  resultant 
on  the  base  makes  with  the  normal  to  the 
base.  This  should  not  exceed  the  angle 
<£'  of  friction  of  masonry  on  earth  or  sliding 
will  occur.  The  factor  of  safety  against 

tan  $' 

sliding  will  be  at  least.  -        -  and  if  pos- 
tan0 

sible  this  factor  should  not  exceed  two. 
The  average  angles  of  friction  of  masonry  on 
dry  clay,  dry  earth  and  firm  sand  or  gravel, 
are  27°,  30°,  35°  respectively,  but  on  wet 
clay,  11°  to  18°  has  been  given.  Hence 
it  is  not  always  possible,  for  reasonable 
thicknesses  of  wall,  to  ensure  a  factor  of 
safety  of  2  against  sliding.  In  such  cases, 
the  base  should  be  inclined,  so  that  the  re- 
sultant on  it,  should  make  an  angle  with 

in  Fig.  11,  the  moment  formula,  deduced  in  a 
similar  manner  to  that  of  Art.  58,  is  as  follows: 


tan2  w-f  |  —4Ki(f  cos  a -sin  a)  -f  tan  /3    tan  u 

=—  2Ki[3c  sec  a +2  tan  a  (/  cos  a -sin  a)] 

+tan  0  (tan  a+tan  0). 

The  formula  is  adapted  to  the  case  where  the  inner 
face  of  the  wall  leans  away  from  the  earth,  by 
replacing  sin  a  by  (—sin  a)  and  tan  a  by  (—tan  a). 


135 

its  normal,  much  less  than  the  probable 
angle  of  friction.  In  the  tabular  thick- 
nesses, no  foundation  slab  was  assumed, 
though  one  is  always  desirable  and  it 
should  be  constructed  with  the  toe  pro- 
jecting beyond  the  front  face  of  the  wall 
sufficiently  to  allow  the  resultant  on  the 
base  to  pass  as  near  its  center  as  is  prac- 
ticable and  thus  distribute  the  pressure 
on  the  base  more  uniformly. 

For  an  actual  wall,  the  unit  pressure  on 
the  base  (the  "  soil  pressure  ")  should  be 
computed  by  (1),  Art.  15  and  if  too  large, 
the  foundation  slab  must  be  widened,  so 
as  not  to  subject  the  soil  to  a  greater 
pressure  than  is  accepted  as  safe. 

If  a  value  of  t  is  desired,  for  a  value  of 
e/w  intermediate  between  f  and  -f,it  can  be 
found  with  substantial  accuracy,  by  ordi- 
nary interpolation  from  the  tabular  values, 
assuming  a  linear  variation. 

68.  On  referring  to  the  column  of 
volumes  (or  areas  of  cross-sections  for 
a  length  of  wall  unity)  it  will  be  observed 
that  for  level-topped  earth,  the  types 
are  economical  in  the  order, 
3,  5,  2,  4,  1. 


13C 


. 

O     0     0     0 

—  1  i-(i-l  (M 

o    o    o    o 

o    o    o    o 

<N        —ICO        CO 
—  I        (Ni-(        <M 

e  •» 

OCOOi  ^H 

Cl  CO  —  (CO 

SSS3 

SS2S 

^g^ggt^ 

•^    '     '^     ' 

•3 

•*  —1  CO  CO 

co  ^  cox 

CO-*  MCD 
OOOO 

S3IS 

CO  C*O  ^t1  CO  CO  *O 

^        ^ 

Xa 

•*  —<  CO  CO 

co  ^  cox 

CO  'O  CO  »O 

xceo  c-i 

O  CM  I-H  <M 

CO  'O  O  O  CO  CO 

- 

33SS6 
coiocoio 

I3S1 

COOt^O 
—  (XCOO 

CO  t^Tf  b-  ?Ti-( 

O  —*  t^'M  COO 
^*  ^f  ^O  ''f  "^  CO 

-l> 

s^5$ 

CO  CO  >O  >O 

co  co  to  >o 

CO       CO»O       tO 

ti 

§^§^ 

ssg? 

CO  CO  CO  CO 

Tf<  X  Tt<  X 

CO        COCO        CO 

Tf            X   •*             X 

•«* 

d 

c8 

co     co 

<N       (M 

=^s 

°C^°<N~ 

CO               CO 

o     \  o     ^^ 

C^              (M 

8 
1 

0 

o 

1 

1 

d 

0 

-to 

-Ico 

O 

§,' 

B 

- 

." 

CO 

- 

1S7 


<£> 

0     O     O     O 

(M  J.  CO  O5 

«[«• 

OiCC  I-H  Oi 

1    +    1    + 

1 

o  ^^  o  o 

TfOCt^  00 

~ 

O5  I-H  Ci  O 
TT  OCI>CO 

- 

Olp*  OiO 

•  I* 

Nw'5'5 

.'* 

oc  10  oc  »o 

O  C^  O  (M 

I 

°§«s 

8 

G 

^H     O 

a 

c3 

-'• 

a 

B 

. 

I    a 

r< 


138 


Types  3  and  5  are  nearly  equal  in 
volume,  but  the  pressure  on  the  base  is 
better  distributed  in  number  5. 

When  the  earth  surface  slopes  at  the 
angle  of  repose,  the  volumes  increase  in 
the  order  3,  2,  4,  5,  1. 

The  value  of  t  is  the  width  of  the  wall  at 
the  base,  tf,  the  width  at  top,  both  for 
h=l.  They  likewise  represent  the  ratios 
t/h,  t'/h  for  any  height  of  wall  h.  Thus 
for  h=W  ft.,  type  2,  i=0,  e/w  =f,  width 
at  base  =  3.46  ft.,  width  at  top  =  1.80  ft. 

69.  Walls  with  projections  at  intervals, 
on  the  exterior  or  interior,  are  known  as 
buttressed  or  counter -forted  walls  respectively. 
Fig.  13  shows  a  good  form  of  buttressed 


BUTTRESS. 


wall,  with  the  face  in  the  form  of  arches, 
convex  from  the  earth  side.     In  designing 


139. 

such  walls,  moments  are  taken  about 
the  outer  toes  of  the  buttresses.  The 
great  objection  to  counterforted  walls  in 
masonry  not  reinforced,  is  that  the  coun- 
terforts are  apt  to  break  away  from  the 
face  wall;  so  that  they  have  not  found 
favor  in  America,  in  spite  of  the  large 
economy  shown.  When  reinforced  con- 
crete is  used,  they  present  a  very  effective 
type  of  wall. 


140 


APPENDIX  I. 


DESIGN  FOB  A  VERY   HIGH   MASONRY 
DAM. 

ENGINEEKS  are  by  no  means  agreed  upon  the 
proper  profile  to  give  high- masonry  dams ;  although 
the  three  conditions,  that  there  shall  be  no  tension 
at  any  horizontal  joint,  safe  unit  stresses  every- 
where, and  no  possible  sliding  along  any  plane  joint, 
seem  to  be  generally  accepted  as  essential  to  a  good 
design. 

The  writer  suggests  one  more  condition,  that 
the  factors  of  safety  against  overturning  about  any 
joint  on  the  outer  face  shall  increase  gradually  as 
we  proceed  upwards  from  the  base,  to  allow  for  the 
proportionately  greater  influence,  on  the  higher 
joints,  of  the  effects  of  wind  and  wave  action,  ice, 
floating  bodies,  dynamite,  or  other  accidental 
forces.  The  exact  amount  of  increase  must  be 
largely  a  matter  of  judgment;  but,  if  the  principle 
is  accepted,  it  can  only  resul'  in  making  stromger 
dams. 


141 

The  accompanying  sketch,  cf  a  dam  258  feet 
high  to  the  surface  of  water  (see  also  ' '  Engineering 
News"  for  June  23,  1888)  satisfies  the  four  condi- 


258 


tions  named,  and  will  be  briefly  described.  The 
dam  is  of  the  same  total  height  (265  feet)  and 
volume  (nearly)  as  the  proposed  Quaker-Bridge 
dam,  and,  for  ease  of  comparison,  is  designed,  as 


142 

was  that  dam,  for  masonry  weighing  2|  times  as 
much  as  water.  The  dam  is  24  feet  wide  at  top, 
38  feet  wide,  50  feet  below  the  surface  of  water  (7 
feet  below  the  top),  and  196.1  feet  wide  at  the 
base.  The  up-stream  face  is  vertical  for  the  first 
57  feet  from  the  top,  and  then  batters  at  the  rate 
of  30  feet  in  200  to  the  base.  The  outer  face  slopes 
uniformly  from  the  top  to  50  feet  below  the  water 
surface,  and  then  slopes  uniformly  to  the  base. 

The  curves  of  pressure,  for  reservoir  full  or 
empty  (the  lines  connecting  the  centres  of  pressure 
on  the  different  horizon tol  joints  are  here  styled 
the  curves  of  pressure),  are  found  as  hitherto  ex- 
plained, and  are  seen  to  lie  well  within  the  middle 
third  of  the  base,  so  that  the  horizontal  joints  under 
the  static  pressure  are  only  subjected  to  compres- 
sion throughout  their  whole  extent.  Further,  it 
was  found  by  construction,  that  if  a  horizontal 
force  be  assumed  as  acting  at  the  surface  of  water, 
of  such  intensity  (29,375  pounds)  as  to  cause  the 
total  resultant,  on  the  joint  50  feet  below  the  water 
level,  to  cut  the  joint  one-third  of  its  width  from 
the  outer  face ;  then  if  this  same  force,  acting  at 
the  surface  of  water,  is  combined  in  turn  with  each 
of  the  other  resultants  on  the  lower  horizontal 
joints,  the  new  centres  of  pressure  will  still  lie  well 
within  the  middle  third  for  the  lower  joints.  To 
secure  uniformity  of  results  for  all  the  joints,  the 
width  at  the  50  feet  level  should  be  increased, 
although  it  is  now  much  greater  than  ordinarily 


constructed.  If,  however,  the  effects  of  earthquake 
vibrations  are  to  be  guarded  against,  we  cannot  re- 
place them  by  the  action  of  a  single  force  acting  at 
the  surface,  so  that  the  increased  width  of  the 
upper  joints  must  be  largely  a  matter  of  judgment. l 
The  numbers  to  the  right  of  the  figure,  in  the 
form  of  a  fraction,  give  for  the  corresponding 
joints,  for  the  upper  numbers,  the  factor  against 
overturning,  or  the  factor  by  which  it  is  necessary 
to  multiply  the  static  horizontal  thrust  of  the  water 
to  cause  the  total  resultant  to  pass  through  the 
outer  edge  of  the  joint  considered;  and  for  the 
lower  numbers,  the  ratio  of  the  weight  of  masonry 
above  a  joint  to  the  static  thrust  of  water  against 
it ;  which  is,  in  a  certain  sense,  a  factor  of  safety 
against  sliding  on  a  horizontal  joint.  These  factors 
are  seen  to  increase  from  the  base  upwards,  so  that 
the  suggested  fourth  condition  is  satisfied. 

i  It  is  stated  in  Engineering  News  for  June  30,  1888,  on 
the  authority  of  Mr.  Thomas  C.  Reefer,  President  Ameri- 
can Society  of  Civil  Engineers,  that  "an  ice  bridge  of 
about  90  feet  span,  between  *wo  fixed  abutments,  ex- 
panded so  from  a  rise  of  temperature,  as  to  rise  3  feet  in 
the  centre."  If  we  regard  the  arch  thus  formed  as  free 
to  turn  atthe  abutments  and  at  the  crown,  we  easily  find 
for  ice  one  foot  thick,  the  horizontal  thrust  H  exerted  at 

the  abutments,  from  the  equation,  3H=  ®^  x  ^,  to  be 

in  pounds  per  square  foot  H=  21,094  pounds.  Much 
higher  pressures  may  possibly  be  expeiienced  sometimes 
near  the  top  of  high  dams  in  northern  latitudes,  and  it 
seems  only  proper  to  include  such  contingencies  in  their 
design. 


144 


The  unit  stresses,  in  pounds  per  square  foot,  at 
the  outar  edgas  of  tha  joints  for  reservoir  full,  and 
at  the  inner  edges  for  reservoir  empty,  are  given 
in  columns  4  and  5  of  the  following  table,  being 
computed  from  the  formula 


Depth  of 
Joint  below 

Water 

Weights 

Pressure 

Pressure 

Water 
Level. 

Pressure. 

of 
Masonry. 

at 
outer  edge. 

at 
inner  edge. 

1. 

2. 

3. 

4. 

5. 

feet. 

Ibs. 

Ibs. 

50 

1,250 

4,417 

8,860 

10,460 

100 

5,000 

11,540 

13,480 

16.130 

150 

11,250 

23,4*0 

20,410 

21,440 

200 

20,000 

40,040 

27,330 

27,170 

250 

31,250 

61,4-20 

34,350 

33,130 

258 

33,282 

65,270 

35,300 

34.120 

The  numbers  of  columns  2  and  3  for  one  foot  in 
length  of  the  wall  are  expressed  in  weights  of  cubic 
feet  of  water,  and  must  be  multiplied  by  62.5  to 
reduce  to  pounds. 

The  unit  pressures,  although  necessarily  high, 
are  still  permissible.  By  spreading  the  lower  part 
of  the  dam  still  more,  these  unit  stresses  would  be 
theoretically  diminished,  though  it  is  likely  that  in 
reality  the  pressures  ab  the  positions  of  the  old  toes 
would  not  be  very  materially  altered ;  but  the 
masonry  being  surrounded  with  other  masonry 
could,  most  probably,  stand  a  higher  pressure. 


The  unit  pressures  p  given  in  columns  4  and  5 
are  not  the  maximum  normal  pressures  at  the 
faces.  In  Appendix  III  (e),  it  is  proved  that  the 
maximum  normal  stress  at  a  face  acts  parallel  to 
that  face  on  a  plane  at  right  angles  to  it  and  that 
its  intensity  is  given  by  the  formula,  f=psec2<f» 
where  <£  is  the  angle  the  face  makes  with  the 
vertical.  In  this  example,  where  <£=31°23'  for 
the  outer  face  and  8°  32'  for  the  inner  face,  the 


FIG.  15. 

values  of  /  at  the  outer  and  inner  faces  are  found 
by  multiplying  the  numbers  given  in  columns  4 
and  5  by  1.37  and  1.02  respectively. 

The  first  derivation  of  the  important  formula, 
/=psec2<£,  has  been  credited  to  Levy  by  Dr. 
Unwin,1  who  likewise  states  that  in  several  old 
dams  which  have  lasted  for  centuries,  the  values 
of  p,  ranged  from  11£  to  14  tons  per  square  foot, 
giving  the  maximum  compressive  stresses  /  from 
15  to  20  tons  per  square  foot  (234  to  311  Ibs.  per 
square  inch). 

1  Minutes  of  Proceedings,  Inst.  C.  E.,  Vol.  CLXXII. 
Part  II,  p.  134. 


146 

The  so-called  factors  against  overturning  are  not 
true  ones,  for  a  computation  shows  that  if  the 
water  pressure  down  lo  the  joints  50,  100,  and  150 
feet  below  the  surface  should  become  2,  1|,  1^ 
times  the  original,  respectively,  that  tension  would 
just  begin  to  be  exerted  at  the  inner  face.  This 
would  happen  for  lower  joints  for  thrusts  about  1£ 
to  11  times  the  original.  If,  from  any  cause,  as 
accidental  forces  at  the  top,  earthquakes,  etc. ,  the 
thrusts  should  be  increased  over  these  amounts, 
causing  tension  at  the  inner  edges  beyond  the 
capacity  of  the  mortar  to  withstand,  (he  joints 
would  crack  and  open,  water  would  get  in,  dimin- 
ishing the  weight  of  the  masonry  materially,  the 
centres  of  pressure  would  move  outwards,  and  the 
unit  pressures  at  the  outer  toes  would  very  much 
increase,  leading  perhaps  ultimately  to  the  destruc- 
tion of  the  dam  through  sliding,  overturning,  or 
crushing  at  the  down -stream  face. 

We  shall  now  consider  the  capacity  of  resistance 
of  the  dam  to  sliding  along  any  oblique  joint  as 
AK. x  Let  AB  represent,  in  magnitude  and  direc- 
tion, the  resultant  of  the  water  pressure  and  weight 
of  masonry  on  the  horizontal  joint  AH,  and  let  the 
vertical  AD  represent  the  weight  of  the  triangular 
mass  AHK,  all  for  one  foot  in  length  of  the  wall. 
Draw  DN  j_  AK  and  BN  \\  AK  to  intersection  N; 

1  See  Annales  des  Fonts  et  Chausse'es  for  May,  1887. 


tlien  DN  =  component  of  BD  normal  to  plane 
AK,  and  DN  X  tan  <f>  ( where  tan  0  =  co- efficient 
of  friction  of  masonry  on  masonry)  is  the  total 
friction  that  can  be  exerted  by  the  plane  AK.  If 
we  lay  off  angle  NDE  =  <t>  (taken  as  35°  here)  to 
intersection  E  with  the  parallel  component  BN, 
we  have  DN  tan  <t>  =  EN,  so  that  BE  must  be 
resisted  by  cohesion ;  and  the  unit-shearing  stress 

7?  W 

along  the  plane  AK  =  — _  .     If,  now,  we  produce 
AIL 

KE  on  to  intersection  C,  with  AB  produced,  we 

BC 

have  the  unit  shear  represented  by ,  which  is  a 

AL/ 

maximum,  for  various  planes  passing  through  A, 
when  C  is  farthest  removed  from  B. 

On  effecting  this  construction,  then,  for  a  series 
of  planes  passing  through  A,  we  quickly  find  the 
plane  which  will  have  to  supply  the  maximum  in- 
tensity of  shear,  or  the  plane  of  rupture,  to  lie  near 
AK  (there  is  very  little  difference  for  a  series  of 
planes  lying  near  each  other) ;  and  the  shear  per 
square  foot  required  to  resist  sliding,  in  addition  to 
the  frictional  resistance,  to  be  about  twenty-seven 
hundred  and  fifty  pounds.  To  offer  the  greatest 
resistance  to  sliding,  there  should  be  no  regular 
courses,  and  the  stones  should  break  joint  verti- 
cally as  well  as  horizontally,  or  the  courses  near  the 
outer  face  should  be  curved  so  as  to  be  approxi- 
mately normal  to  that  face.  For  a  retaining-wall 
of  dry  rubble,  carelessly  laid,  we  see  that  there  is 
every  probability  of  failure  by  sliding  along  some 
inclined  plane.  Here  the  stones  must  be  carefully 


148 

interlocked  to  prevent  sliding.  For  the  reservoir- 
wall,  where  the  best  cement  is  used,  and  the  joints 
are  broken,  there  should  be  no  fear  of  sliding  when 
sufficient  thickness  is  given  to  avoid  tension.  In 
the  Habra  dam,  a  hundred  and  sixteen  feet  high, 
this  was  not  done ;  and  the  dam  broke  along  a 
plane,  passing  through  the  outer  toe  nearly,  and 
making  the  angle  of  friction  <f>  of  masonry  on 
masonry  with  the  horizontal. 

It  is  well  to  note,  tco,  that  friction  alone  will  not 
prevent  sliding  along  planes  inclined  not  far  from 
the  horizontal  as  well  as  those  below,  so  that  a 
proper  resistance  to  shear  must  be  provided  for  in 
every  dam.  Possibly  the  weak  point  of  many  dams 
is  in  this  very  particular. 

The  capacity  of  the  dam  in  question  to  resist 
rotation  about  the  toe  of  an  inclined  base  may  next 
be  tried,  and  it  will  be  found  to  be  stable ;  for  the 
weight  of  masonry,  as  well  as  its  arm,"  increases  to 
counterbalance  the  increase  of  arm  of  the  water- 
thrust.  The  dam  thus  satisfies  all  the  conditions 
of  stability ;  and,  although  some  of  its  dimensions 
may  be  changed  with  advantage  perhaps,  it  yet 
suffices  very  well  to  point  out  the  principles  of 
design. 

See  Engineering  News  for  January  12,  1893  and 
May  9,  1907  for  effects  of  expansion  of  ice. 


APPENDIX  II. 


STRESSES    IN    MASONRY    DAMS.1 

THE  object  of  this  investigation  is  to  deter- 
mine the  amounts  and  distribution  of  the  stresses 
in  a  masonry  dam,  at  points  not  too  near  the 
foundations,  having  assumed  the  usual  "  law  of 
the  trapezoid,"  that  vertical  unit  pressures  on 
horizontal  planes  vary  uniformly  from  face  to 
face. 

Experiment  indicates  that  such  vertical  stresses 
increase  pretty  regularly  in  going  from  the  inner 
to  the  outer  face,  for  reservoir  full,  until  we  near 
the  down-stream  or  outer  face,  where  the  stress 
gradually  changes  to  a  decreasing  one,  which 
decrease  continues  to  the  end  of  the  horizontal 
rection.  The  law  of  the  trapezoid  is  thus  only 
approximately  true  over  part  of  the  section,  but, 
as  it  gives  an  excess  pressure  where  it  attains  a 
maximum,  it  errs  on  the  safe  side. 


1  What  follows  in  Appendices  II  and  III  was  first 
given  by  the  author  in  Trans.  Am.  Soc.  C.E.,  Vol. 
LXIV,  p.  208. 


150 

The  profile  of  the  dam  selected  is  of  the  trian- 
gular type,  with  some  additions  at  the  top,  but  the 
method,  used  in  determining  the  stresses  is  general 
and  will  apply  to  any  type  of  profile.  The  final 
equations  will  give,  at  any  (interior  or  exterior) 
point  of  the  horizontal  section  considered,  the 
vertical  unit  stress  on  the  horizontal  section,  the 
normal  stress  on  a  vertical  plane,  and  the  unit 
shear  on  either  horizontal  or  vertical  planes. 
From  these  stresses,  the  maximum  and  minimum 
normal  stresses,  and  the  planes  on  which  they  act, 
can  be  determined,  and  ultimately,  if  desired,  the 
stress  on  any  assumed  plane  can  be  ascertained. 

The  solution  presented  is  approximate,  which  is 
justifiable,  in  view  of  the  approximation  involved 
in  "the  law  of  the  trapezoid"  used.  The  results, 
however,  are  practically  correct,  as  will  be  evident 
from  the  checks  applied,  resulting  from  the  exact 
theory  given  in  Appendix  III.  The  theory  used, 
being  simple,  should  be  easily  followed. 

Let  Fig.  16  represent  a  slice  of  the  dam  con- 
tained between  two  vertical  parallel  planes,  1  ft. 
apart  and  perpendicular  to  the  faces.  The  batter 


of  OB  is  !!?-*!?;    that  of  OE  being  _4_  -          . 
200        1  200         1 

The  batter  of  the  inner  face  was  found  by  trial, 
so  that  the  centers  of  pressure  on  horizontal 
sections,  for  reservoir  empty,  should  nowhere  pass 
more  than  a  fraction  of  a  foot  outside  the  middle 
third  of  the  section.  The  simple  type  of  profile 
shown  was  adopted  for  ease  of  computation. 

For    convenience    in    subsequent    computations, 
the  breadths,  b  =  EB,  of  horizontal  sections,  corre- 


151 

spending  to  various  depths,  h,  below  the  surface 
of  the  water  in  the  reservoir,  are  given,  all  dimen- 
sions being  in  feet: 


/i  =  199.0, 
^==199.5, 
h  =  200.0, 
h  =  200.5, 
h  =  201.0, 


6  =  133.330; 
6  =  133.665; 
6  =  134.000; 
6  =  134.335; 
6  =  134.670. 


Take  the  weight  of   1   cu.  ft.   of  masonry  equal 
to    1;     then    the    weight    of    masonry    above    any 


,  =  a» 


section  is  equal  to  the  corresponding  area  in 
Fig.  16  above  that  section.  The  area  of  the  por- 
tion above  EOB  is  readily  found  to  be  712,  and 
its  moment  about  the  vertical,  AO,  is  11,603,  the 
unit  of  length  being  the  foot.  In  Fig.  16,  D  is 


152 

where  the  vertical  through  the  center  of  gravity 
of  the  dam  above  the  joint,  EB,  cuts  that  joint, 
and  C  is  the  center  of  pressure  on  that  joint  when 
the  water  pressure  on  EO  is  combined  with  the 
weight  of  masonry,  W ,  above  EB. 

As  h  varies,  suppose  each  horizontal  joint,  in 
turn,  marked  similarly  to  the  joint  at  7i  =  200,  with 
the  letters  E,  A,  D,  C,  B;  then,  for  any  joint,  on 
taking  moments  of  the  triangles,  AOB,  AOE,  and 
the  area  above  OB  about  A,  we  find 


—  (AB*-EA  2)+ 11,603 


AD 


Assuming  that  the  masonry  weighs  2^  times  the 
water  per  cubic  unit,  then  the  weight  of  a  cubic 

2 

foot  of  water  is  —  .      It  would  entail  but  little  extra 
5 

trouble  here,  where  the  inner  face  has  a  uniform 
batter  throughout,  to  include  the  vertical  conv 
ponent  of  the  water  pressure  on  the  face,  EO; 
but  it  will  be  neglected,  as  usual. 

The  horizontal  water  pressure  for  the  height,  h, 

is  thus,  _  X  __  =  _  h2,  and   its   moment    about   C   is 
525 

Lh*X--h=  Lh*. 
5  3        15 

Taking  moments  of  W  and  water  pressure  about 
C,  we  have  at  once, 


15     W 


153 

From  the  last  two  formulas,  we  derive  the 
following  results: 

h  W                       AD                     DC 

199  13978.335  40.49141  37.58483 

200  14112.000  40.70316  37.79289 

201  14246.335  40.91488  33.00089 

A  seven-place  logarithmic  table  was  used 
throughout,  the  aim  in  the  computations  being 
to  get  the  seventh  significant  figure  correct  within 
one  or  two  units.  The  necessity  for  this  accuracy 
will  be  seen  later. 

The  distances  EC  and  CB  are  now  readily 
derived. 

ForA  =  199,  #C=82.05624,  C5  =  51.27376; 
A=200,  EC  =  82.49605,  CB  =  51.  50395; 
h  =  201,  EC=  82.93577,  CB=  51.  73423. 

On  any  plane,  EB,  the  vertical  unit  pressure 


b2 

4b 
at  E=pi=  — 


b2 

where  b=EB,  and  W  is  the  weight  of  masonry 
above  the  plane.  This  follows  from  the  assumed 
"law  of  the  trapezoid." 

From  these  formulas  we  derive: 

At  h  =  199,  pi  =  177.45483,  p2  =32.22542  ; 
h  =  200,  pi  =  178.3855,  p2  =32.24139; 
h  =201,  pi  =  179.3160,  p2  =32.25798. 


Call  p  the  vertical  unit  stress  at  a  distance,  x'  '  , 
from  E;  then 

p\~—pi  , 

r 


, 

b 
and  the  total  stress  on  the  base,  x'  ',  is 


(i) 


To  find  the  unit  shear  on  vertical  or  horizontal 
planes,1  consider  a  slice  of  the  dam,   bounded  by 


1  The  writer  desires  here  to  acknowledge  his  indebted- 
ness to  a  recent  paper  on  "Stresses  in  Masonry  Dams," 
by  Ernest  Prescot  Hill,  M.  Inst.  C.E.,  published  in 
Minutes  of  Proceedings,  Inst.  C.E.,  Vol.  CLXXII, 
p.  134.  Mr.  Hill  considers  the  case  of  a  dam  with  a 
vertical  inner  face.  By  the  aid  of  the  calculus,  he 
effects  an  exact  solution,  which  leads  to  general  formulas 
for  shear  and  normal  pressures  on  vertical  planes. 

The  principles  at  the  base  of  his  method,  though 
somewhat  disguised  by  the  calculus  notation,  are  essen- 
tially the  simo  as  those  used  by  the  author. 

Mr.  Hill  ascribes  to  Professor  W.  C.  Unwin  the  sugges- 
tion, "that  the  shearing  stress  at  any  point  may  be 
found  by  considering  the  difference  between  the  total 
net  vertical  reactions  [between  that  point  and  either 
face]  along  two  horizontal  planes  at  unit  distance 
apart,"  and  states  that  Prof.  Unwin  "has  applied 
the  principle  to  a  triangular  dam  by  the  use  of  alge- 
braical methods." 

Dr.  Unwin  states  (Proc.  Inst.  C.E.,  Vol.  CLXXII, 
Part  II,  p.  161)  that  he  ascertained  after  his  papers 
were  written,  that  by  a  different  method,  Levy  had 
previously  arrived  at  the  same  conclusions. 


155 

horizontal  planes  at  ft  =  199  and  A  =200,  the  water 
face  and  a  vertical  plane,  at  a  distance,  x,  from  the 
inner  face  (Fig.  17),  in  equilibrium  under  the  water 
pressure  acting  horizontally  on  its  left  face  and 
the  forces  exerted  by  the  other  parts  of  the  dam 
on  the  slice.  These  forces  consist  of  the  uniformly 
increasing  stress,  P' ,  on  top,  acting  down;  the 
uniformly  increasing  stress,  P,  on  the  bottom, 
acting  up;  a  shear  acting  on  the  vertical  plane 


P' 


FIG.  I/. 


at  the  right,  of  average  intensity  qi  per  square 
foot,  the  weight  of  the  body  (x  —  0.01),  besides  the 
horizontal  forces  to  be  given  later.  The  vertical 
component  of  the  water  pressure  is  here  neglected, 
as  usual.  The  origin  for  x  is  taken,  here  and  in 
all  subsequent  work,  at  the  level,  ft  =200,  at  the 
inner  face. 

For  equilibrium,   the  sum  of  the   vertical  com- 
ponents must  be  zero. 

Therefore, 

5l  =  (x-0.01)4-P'-P.      ...      (2) 


156 

To  find  P',  substitute  in  Equation  (1),  x'  —x 
-0.02,  pt  =32.22542,  pi  -pi  =  145.22941,  k  = 
133.330,  giving  P'  =32.20364*  +  0.5446238**- 
0.6442906.  For  P,  *'=*,  p2  =  32.24139,  y>i-p2  = 
146.1441,  and  6  =  134;  therefore, 

P  =  32.24139* +0.5453138*2. 

Substituting  in  Equation  (2),  we  derive  the 
average  unit  shear, 

g,  =  _  0.6542906  -  0.96225*  -  0.000690D*2.     .      (3) 

This  value  of  q\  is  strictly  correct  w   en  x^_  0.02 
It  is  slightly  in  error  when  0<*<0.02. 


t=200 


x  +0.01 


X  1 


X  +  0.02 


FIG.  18. 


A  similar  investigation  holds  to  obtain  the 
average  unit  shear,  92  (Fig.  18),  on  a  vertical  plane, 
at  a  distance,  *,  from  E,  extending  from  the 
level,  ft  =  200,  to  the  level,  h=20l. 

We  have,  for  equilibrium, 

P".  (4) 


157 

We  find  P"  by  substituting  in  Equation  (1), 
z'  =  (z+0.02),  pz  =32.25798,  pi-p2  =  147.05S02, 
and  6  =  134.67.  P"=32.27982o:  +  0.5459941x2  + 
0.6453780.  Substituting  this,  and  the  value  pre- 
viously found  for  P,  in  Equation  (4),  we  derive, 

q2=  -0.6353780  +0.961 57:c-0.0006803z2.       (5) 

This  is  strictly  correct  only  when  x>_0. 

The  mean,  ^(51+92),  of  these  average  sheare  will 
be  assumed  as  approximately  equal  to  the  inten- 
sity of  shear  at  the  point,  G(x  =  EG),  at  the  level, 
h  =  200.  Call  q  this  intensity  of  shear  on  a  ver- 
tical plane  at  G]  therefore, 

q=  -0.6448343  +0.96191*  -0.0006856*'.       (6) 

Checks. — By  Appendix  HI  (6)  and  (d),  the  exact 
value  of  q,  at  either  face,  =/>  tan  <f>,  where  p  = 
vertical  unit  normal  stress  at  the  face  and  ^  is 
the  angle  the  face  makes  with  the  vertical.  Thus, 
at  the  inner  face,g=  -32.24139X0.02  =  -0.6448278, 
whereas  Equation  (6)  gives  for  x  =  0,  q=  — 
0.6448343. 

At  the  outer  face,  the  exact  value  is,  178.3855 
X0.65  =  115. 9506,  whereas  Equation  (6)  gives,  for 
a;  =  134,  q  =  115.9405. 

A  still  more  searching  test  can  be  devised.  It 
is  a  well-known  principle  that  the  intensity  of 
shear  at  a  point,  on  vertical  or  horizontal  planes, 
is  the  same  [Appendix  III  (a)].  Therefore,  regard- 
ing Equation  (6)  as  giving  the  horizontal  unit 


158 

shear,  at  the  level,  h  =  200,  where  b  =  134  ft.  ;    the 
total  shear,  from  face  to  face,  on  this  level,  is 


/*: 
I 

Jx 


:r=134 


This  should  equal  the  total  water  pressure  down 

to    the   same    level,    -=-  (200)  2=  8000.     Formula  (6) 
o 

thus  gives  practically  exact  results. 

In  order  to  find  the  normal  unit  stress  on  a 
vertical  plane,  we  shall  assume  that  q\,  given  by 
Equation  (3),  equals  the  intensity  of  shear  on  a 
vertical  or  horizontal  plane  at  the  point,  x,  at 
h  =  199.  5;  and  that  qz,  given  by  Equation  (5), 
gives  the  shear  intensity  at  x  at  h  =200.5.  This 
evidently  supposes  that  the  shear  intensity  in- 
vreases  uniformly,  vertically,  from  h  =  199  to 
fc  =  201. 

Consider  a  portion  of  the  dam,  Fig.  19,  bounded 
by  the  water  face;  the  plane,  FM,  at  the  level, 
h  =  199.5,  on  which  the  total  shear  is  Q',  the  plane 
EN,  at  the  level  200-5,  on  which  the  total  shear 
is  Q,  and  the  vertical  plane,  MN,  1  sq.  ft.  in  area, 
on  which  the  average  normal  stress  is  p'  '.  The 
water  pressure  on  EF  will  be  supposed  to  be 
exerted  horizontally.  It  is  equal  to  80  units. 
Assuming,  as  stated,  that  91  =  intensity  of  hori- 
zontal shear  at  M,  and  92  =  the  corresponding 
intensity  at  N,  we  have,  taking  the  origin  as 
before  at  O, 


rx  rx 

'=    I     qidx;     Q=   I     qi  dx; 

Jo.oi  J-o.oi 


159 


or, 

Q'=  0.006494794  -0.6542906z  +  0.481 125*2 

-0.00023z»; 

Q  =  _  0.00640186  -  0.6353780x  +  0 .480785*' 

x8 

-  0.0006803  o". 
o 

Checks. — The  total  water  pressure  for  h  =  199.5 

is  -^(199.5) 2  =  7960.05   and   for  h  =  200.5,  -^-(200.5)2 
o  o 

=  8040.05.     The    first    should    equal    Q',    for    x  = 
133.665,  or  7959.22;     the  second  should  equal  Q, 

Q' 


h  =  199.J 
200    —  ^ 
=  200.5 

F 

an—  0.0? 

X 

1 

r 

si  

x  +  0.01 

1 

FIG.  19. 

for  x  =  134.335,  or  8041.12.  The  slight  differences 
tend  to  give  confidence  in  the  results. 

For  equilibrium,  the  sum  of  the  horizontal  forces 
acting  on  EFMN,  Fig.  19, must  be  zero;   therefore, 

p'=80+Q'-0,      ....     (7) 
p'  =  80.01  -0.0189*  +  0.00034*2_o.00000323x». 

This  average  stress  will  now  be  assumed  to  be  the 
intensity  of  the  horizontal  unit  stress  on  vertical 
planes  at  h=200. 


160 

It  will  now  be  perceived  why  a  seven-place  table 
was  necessary  in  the  computations,  the  coefficients 
of  x2  and  x3  having  only  two  or  three  significant 
figures  in  the  final  result.  If  the  planes  originally 
had  been  taken  0.1  ft.  apart  vertically,  a  ten-place 
table  would  have  been  required. 

Checks.— The  value  of  p' ',  for  x  =  0,  p'  =80.012896, 
is  the  same  as  that  given  by  Appendix  III  (d), 
80+0.6448X0.02.  When  z  =  134,  the  formula 
gives  p' =  75.81,  whereas  the  exact  theory,  Appen- 
dix III  (6),  gives  p'=m2p  =  (0.65)2  XI 78.39  =  75.37. 
The  difference  is  0.44  at  the  outer  face.  For  any 
other  point,  it  might  be  assumed  to  vary  with  x, 
so  that  it  could  be  corrected  by  substracting 

0  44 

-^-rx=0.0033x  from  the  value  of  p'  above.     For 

JL«54 

ease  of  computation,  the  formula  will  be  written, 
7/  =  80.01  -0.02z  + 0.00034x2 -0.0000l|-.       (g) 

The  first  coefficient  of  x3  cannot  be  counted  on  to 
the  last  two  figures,  hence  we  are  permitted  to 
change  323  to  333  in  that  coefficient.  When 
z  =  134,  Equation  (8)  gives  p' =  75.41,  nearly  the 
exact  value. 

The  three  formulas  for  p,  q,  and  p',  at  the 
level  ^  =  200,  are  thus  as  follows: 

p  =32.24  +  1.09063z; ; 
q=  -0.64+ 0.962* -0.000686Z2; 
p'  =  80.01  -  0.02z +0.00034*2  _  o.OOOOl    . 


1G1 


Since  the  weight  per  cubic  foot  of  masonry  was 
assumed  as  two  and  one-half  times  that  of  water, 
we  must  multiply  the  stresses  given  in  Table  I 

by  ~n  (62.5)  =  156.25,  to  reduce  to  pounds  per  square 

foot;  or  by  1.085,  to  reduce  to  pounds  per  square 
inch. 

TABLE  I. 


X 

0 

10 

25 

50 

p  

32.24 

43.15 

59.50 

86.77 

g  

-0.64 

8.91 

22.98 

45.75 

Max.  /  !  '. 

80.01 
80.02 

80.02 
82.06 

79.66 
94.67 

79.11 
128  .  85 

Min.  /  

32.23 

41  .11 

44.48 

37.03 

8  for  max.  /  .  . 

90°  46' 

77°  06' 

56°  50' 

42°  36 

X 

75 

100 

134 

P  

114.04 

141.30 

178.39 

q  

67  .  65 

88.70 

115.95 

p'  

79.01 

78.08 

75.37 

Max.  /  
Min./  

166.40 
26.64 

203  .  85 
15.52 

253.71 
0 

B  for  max.  /  .  . 

37°  44' 

35°  12' 

33°  01' 

In  Table  1  the  stresses  are  those  experienced  at 
the  level,  h  =  200. 

p  =  vertical  unit  stress  on  a  horizontal  plane; 
q  =  shearing  unit  stress  on  horizontal  or  ver- 
tical planes; 

p'=  horizontal  unit  stress  on  vertical  planes; 
Max.  /=  maximum  normal  stress  acting  on  a  plane 
inclined  to  the  horizontal  at  the  angle, 
S,  given  on  the  last  line; 


102 

Min.  /=  minimum  normal  stress  acting  on  a  plane 
perpendicular  to  the  last. 

From  max.  /  and  min.  /,  with  6,  the  ellipse  of 
stress  can  be  drawn,  and  the  stress  in  any  direc- 
tion, with  the  plane  on  which  it  acts,  can  be 
ascertained. 

It  will  be  observed  that  there  is  no  tension 
exerted  anywhere,  and  that  the  maximum  com- 
pression is  253.71,  or  275  Ibs.  per  square  inch, 
which  is  exerted  at  the  outer  face,  parallel  to  that 
face,  upon  a  plane  at  right  angles  to  the  face. 

In  Appendix  III  (e),  the  important  formula,  for 
the  maximum  normal  intensity  at  the  outer  face, 
acting  parallel  to  that  face, 


is  proved.     In   this  'instance,    p  =  178.39,   tan  <£  = 
0.65,  therefore  0  =  33°  01',  whence  /=253.  71. 

This'  stress  is  unaccompanied  with  any  conju- 
gate stress,  perpendicular  to  the  face.  In  the 
interior  of  the  dam,  where  conjugate  stresses 
prevail,  the  masonry  is  perhaps  better  able  to 
withstand  a  certain  compressive  stress  than  at  the 
face.  The  distribution  of  stresses,  at  the  level, 
h  =  200,  is  shown  in  Fig.  20,  on  the  supposition 
that  the  base  of  the  dam  is  a  little  below  that 
level.  The  connection  with  the  foundation  mate- 
rially modifies  this  distribution;  but  Fig.  20  shows 
the  distribution  for  sections,  say,  from  10  to  20  ft. 
above  the  base,  up  to  the  level  h  =  100,  fairly  well, 
on  the  basis  of  the  trapezoid  law.  As  has  been 


163 

mentioned  before,  this  law  gives  a  pressure  greater 
than  the  actual  at  the  outer  face. 

Since  the  batter  of  the  inner  face  is  very  small, 
the  results  of  Table  I  should  agree  approximately, 
except  near  the  inner  face,  with  those  found  by 
Mr.  Hill  in  the  paper  referred  to  in  the  foot  note. 


-rfrr 

r~n~ 

TT 

mTTTT 

—  1-= 

=dd 

V 

-r^r 

-rrrrrr 

i 

1     !  1 

i  i 

i  i 

i  i  |  i  i  i 

r^K 

/  \ 

FIG.  20. 

Substituting  numerical  values,  Mr.  Hill's  formulas, 
for  h  =200,  reduce  to 

q  =  0.9426x  -  0.0005768x2, 
p'=80  — 0.0001289x2  — 0.0000009615x3; 
giving: 


104: 


X 

0 

10 

25 

50 

75 

100 

134 

q 

0 

9  36  1  23.2) 

45.69 

67.45 

88.49 

115.95 

P' 

80 

79.99 

79.93 

79.56 

78.87 

77.75 

75.38 

On  comparnig  these  formulas  with  those  of  the 
writer,  it  will  be  observed  that  the  absolute  term 
in  the  value  of  q  and  a  consequent  term  of  the 
first  degree  in  x,  in  the  value  of  p' ',  are  lacking 
in  Mr.  Hill's  formulas.  This  results  from  taking 
the  inner  face  as  vertical.  Although  the  coeffi- 
cients also  differ,  it  is  seen  that  the  numerical 
values  are  very  nearly  the  same. 

In  Fig.  21  are  shown,  on  a  drawing  of  the  dam, 
to  scale,  the  lines  of  the  centers  of  pressure  for 
reservoir  full  and  empty. 

To  the  right,  and  under  the  word  "factors," 
are  certain  numbers,  written  in  the  form  of  frac- 
tions. For  any  joint,  the  upper  number  gives 
the  factor  against  overturning,  or  the  number  by 
which  it  is  necessary  to  multiply  the  water  pressure 
down  to  the  joint,  to  cause  the  total  resultant  to 
pass  through  the  outer  edge  of  the  joint  con- 
sidered. The  lower  numbers  give  the  ratio  of  the 
weight  of  masonry  above  a  joint  to  the  water 
pressure  corresponding. 

It  is  believed  that  these  "factors"  should  in- 
crease from  the  base  upward,  to  allow  somewhat 
for  earthquakes,  expansion  of  ice  in  freezing,  etc., 
since  the  effects  of  such  accidental  forces  is  pro- 
portionately greater  on  the  upper  joints. 

Stresses  due  to  water  infiltration  are  not  included 


105 


here;     neither    are    stresses    due    to    temperature 
changes. 

The  unit  stresses,  /,  in  pounds  per  square  inch, 
acting  parallel  to  the  adjacent  face,  are  as  follows, 


FIG.  21. 

and    refer   to    the   outer    edges  of    the    joints,    for 

reservoir  full,  and  to  the  inner  edges   for   reservoir 

empty : 

h  f  at  Outer  Edge,       at  Inner  Edge. 

50  85  58 

100  136  133 

150  204  180 

200  275  228 


166 

The  stresses,  /,  are  normal  pressures  on  planes 
perpendicular  to  the  respective  faces,  and  are  the 
greatest  stresses  that  can  be  experienced  in  the 
dam.  In  fact,  they  are  greater  than  the  true 
stresses,  since  the  trapezoid  law  is  not  exact, 
particularly  near  the  base,  as  before  remarked. 
It  would  then  seem  that  the  dam,  thus  far,  is  safe, 
since  the  maximum  unit  stress  is  less  than  con- 
crete, even,  is  subjected  to  daily,  in  good  practice. 

For  an  actual  construction,  the  outer  face  should 
be  curved,  from  near  h  =  50  to  the  top,  as  shown 
by  the  curved  dotted  line  in  Fig.  21. 

The  subject  of  the  stresses  in  masonry  dams 
has  caused  a  great  deal  of  discussion  among 
British  engineers  in  the  last  two  or  three  years. 
The  subject  was  reopened  by  Mr.  L.  W.  Atcherly 
and  Professor  Karl  Pearson,1  who  gave  the  results 
of  certain  experiments  which  seemed  to  indicate 
considerable  tension  across  vertical  planes  near  the 
outer  toe.  The  late  Sir  Benjamin  Baker,  Hon. 
M.  Am.  Soc.  C.  E.,  also  published  2  the  results  of 
experiments  on  a  model  dam  of  stiff  jelly,  and 
very  recently,  the  "Experimental  Investigations" 
of  Sir  J.  W.  Ottley  and  Mr.  A.  W.  Brightmore  3  on 
elastic  dams  of  "plasticine"  (a  kind  of  modeling 
clay)  and  the  experiments  of  Messrs.  J.  S.  Wilson 
and  W.  Gore4  on  "India  Rubber  Models"  have 
been  presented. 

VMinutes  of  Proceedings,  Inst.  C.  E.,  Vol.  CLXII,  p. 
456. 

2  Ibid.,  Vol.  CLXII,  p.  123. 

3  Ibid.,  Vol.  CLXXII,  p.  89. 

<  Ibid.,  Vol.  CLXXII,  p.  107. 


1GT 


It  is  not  the  object  of  this  paper  to  discuss  these 
later  experiments;  but  it  may  be  remarked  that 
they  show  very  plainly  that  no  tension  exists  near 
the  outer  toe,  but  that  tension  does  exist  at 
the  inner  toe,  where  the  dam  is  joined  to  the 
foundation,  and  it  has  become  a.  serious  matter 
how  to  deal  with  it.  The  influence  of  the  founda- 
tion in  modifying  the  distribution  of  the  stresses 
at  the  base  of  the  dam  was  found  to  be  very  great, 
causing  the  shear  there  to  be  more  uniform  than 
higher  up,  where  the  parabolic  law,  nearly  as  given 
by  the  formulas  above,  was  found  to  hold.  Also, 
above  some  undertermined  plane,  a  small  distance 
above  the  base,  the  usual  "law  of  the  trapezoid" 
was  found  to  be  approximately  correct,  leading  to 
stresses  on  the  safe  side  at  the  outer  toe.  This 
law  leads  to  stresses  at  the  outer  toe  of  the  base 
considerably  in  excess  of  the  true  ones. 

It  was  found,  from  the  rubber  models  particu- 
larly, as  theory  indicates,  that  the  greatest  normal  ' 
pressures    are    exerted    at    the    down-stream    face, 
for    reservoir    full,    and    they    act    in    a    direction 
parallel  to  that  face. 


1G8 


APPENDIX   III. 


RELATIONS  BETWEEN   STRESSED    AT   ANY 
POINT    OF    A    DAM. 

(a)  Consider  a  cube  of  masonry,  Fig.  22,  the 
edge  of  which  has  the  length,  a,  bounded  by  ver- 
tical and  horizontal 
planes  and  subjected  to 
normal  and  shearing 
forces,  caused  by  the 
action  of  the  other  parts 
of  the  dam.  Since  a 
will  be  supposed  to  di- 
minish indefinitely,  the 
weight  of  the  cube, 
which  is  proportional  to 
a3,  is  an  infinitesimal  of 
the  third  order,  and  can 

be  neglected  in  comparison  with  the  normal  forces, 
which  vary  as  a2  and  are  thus  of  the  second  order. 
Similarly,  the  average  unit  stresses  exerted  on 
the  faces  can  be  treated  from  the  first  as  the  unit 
stresses  at  any  point,  A,  of  tho  cube.  As  a 
diminishes  indefinitely,  the  oppositely  directed 


169 


normal  forces  approach  equality  and  balance 
independently;  hence  the  couples  formed  by  the 
shears  on  opposite  faces  must  likewise  approach 
equality;  the  one  being  right-handed,  the  other 
left-handed;  therefore  qaXa  =  q'aXa,  or  q=q'; 
hence,  the  intensities  of  shear  at  a  point  on  two 
planes  at  right  angles  are  equal.  The  relative 
directions  of  the  shears  on  two  planes  at  right 
angles  are  determined,  as  above,  from  the  con- 
sideration that  one  resulting  couple  must  be  right- 
handed  and  the  other  left-handed.  This  applies 
also  to  Figs.  23  to  26. 


FIG.  23. 

(b)  In  Fig.  23,  ABC  is  the  right   section  of  a 
prism  at  the  outer  face,  with  lateral  faces  one  unit 
in  length,  perpendicular  to  the  plane  of  the  paper. 
Let  AB  be  vertical;  tan  <j>  =  m,  a  constant; 

p=  normal  intensity  on  a  horizontal  plane  at 

C; 

p'  =  normal  intensity  on  a  vertical  plane  at  C; 
g  =  shear  intensity  on  horizontal  or  vertical 
planes  at  C. 


170 

The  weight  of  the  prism  is  %ab. 

Balancing  vertical  as  well  as  horizontal  com- 
ponents, we  have,  when  a=AB  and  b  =  AC  are 
very  small, 

pb  =  qa  +  %ab,  nearly; 

p'a  =  qb. 

Dividing  the  first  equation  by  6,  the  second  by 
a,  the  limit,  as  a  and  6  approach  zero,  gives  exactly, 

p  =  q  cot  <f>,     therefore     q  =  mp; 
p'=qtan<j>,     therefore     p'  =  m2p,      pp'  =  q~. 

These  equations  give  the  relations  between  p, 
q,  and  p'  at  the  outer  face.  The  same  relations 
hold  at  the  inner  face,  for  reservoir  empty,  on 
replacing  $  by  0',  the  angle  the  inner  face  makes 
with  the  vertical. 

For  the  remaining  cases,  the  final  limits  will  be 
written  at  once,  since  the  complete  process  of 
deriving  them  is  evident  from  the  above.  In  fact, 
the  weight  of  the  prism,  %ob,  being  of  the  second 
order,  can  be  neglected  in  comparison,  with 
qa,  etc. 

2 

(c)  For  reservoir  full,   calling  iv=  —  h,  the  inten- 

o 

sity  of  water  pressure,  horizontally  or  vertically, 
at  C,  we  have  at  the  inner  face,  putting  tan  $'  =n, 
Fig.  24, 


therefore  p=—  q  +  w; 


171 


(d)  If    the    vertical    component    of    the    water 
pressure  is  neglected,  these  equations  reduce  to 


therefore 


(e)  Since  the  shear  on  the  outer  face  is  zerc, 
therefore,  by  (a),  the  shear  on  a  plane,  AD,  Fig.  25, 
perpendicular  to  the  outer  face,  is  also  zero,  or 
the  stress  on  AD  is  normal. 

Call  /  the  intensity  of  such  a  stress  at  C.  The 
total  pressure  on  AZ)=/X  AD=fb  cos  <f>,  and  its 
vertical  component  is  fb  cos2  <j>,  therefore  balancing 
the  vertical  components, 

pb  =*fb 


therefore 


172 

This  is  a  most  important  formula  for  finding 
the  maximum  normal  intensity  at  the  outer  fact-. 
It  applies  equally  to  the  inner  face  for  reservoir 
empty,  on  changing  $  to  <£',  the  angle  the  inner 
face  makes  with  the  vertical.  For  either  face,  p 
is  the  vertical  normal  unit  stress  at  the  face  con- 
sidered. 

(/)  Principal  Normal  Stresses  at  Any  Point  in 
the  Dam  and  the  Planes  on  which  they  Act. — In  the 
prism,  ABC,  Fig,  26,  let  AB  be  one  of  the  planes 


FIG.  26. 


on  which  the  stress  is  normal.  Let  /  be  its  inten- 
sity. The  stress  on  the  plane,  AB,  of  unit  length 
perpendicular  to  the  plane  of  the  paper,  is  thus 
fc\  its  vertical  component  is  /c  cos  0  =/&,  and  its 
horizontal  component  is  fc  sin  B  -=/a,  8  being  the 
angle  that  A  B  makes  with  the  horizontal 

Place  the  sum  of  the  vertical  forces  acting  on 
ABC  equal  to  zero;  also  place  the  sum  of  hori- 
zontal forces  equal  to  zero. 


173 

fb  =  pb  +  qa,      therefore     /—  p  =  q  tan  0, 
fa  =  qb  +  p'a,     therefore     /—  p'  =  q  cot  8  , 

The  difference  of  the  last  two  equations  gives 

1  —  tan2  e 


The  angles,  0  (differing  by  90°),  computed  from 
this  equation,  give  the  directions  of  the  planes, 
AB,  on  which  the  stress  is  entirely  normal. 

From  an  equation  above,  we  likewise  have 


f-P 

tan  8  = — -. 
2 


This  gives  directly  the  plane  on  which  a  given 
/  acts. 

To  deduce  a  formula  for  /,  take  the  product  of 
two  equations  above: 


±  V(p  +p')2~4(ppf - 


This  equation  gives  the  two  values  of  /  corre- 
sponding   to    the    two    planes    mentioned;      com- 


174 

pressive  when  /  is  positive,  tensile  when  negative 
There  can  be  no  tension  when  pp'  ">_  qz. 
A  better  form  for  computation  is, 


THE  VAN   NOSTRAND   SCIENCE    SEKIES 


No.  47.    LINKAGES:    THE  DIFFERENT  FORMS 

and  Uses  of  Articulated  Links.     By  J.  D.  C.  De  Roos. 

No.  48.    THEORY     OF     SOLID     AND     BRACED 

Elastic  Arches.  .By  William  Cain,  C.E.  Second  edi- 
tion, revised  and  enlarged. 

No.  49.    MOTION  OF  A  SOLID  IN  A  FLUID.     By 

Thomas  Craig,  Ph.D. 

No.  50.    DWELLING-HOUSES;      THEIR     SANT- 

tary  Construction  and  Arrangements.  By  Prof.  W.  Hi. 
Corfield. 

No.  51.    THE  TELESCOPE:    OPTICAL  PRINCI-. 

pies  Involved  in  the  Construction  of  Refracting  and 
Reflecting  Telescopes,  with  a  new  chapter  on  the 
Evolution  of  the  Modern  Telescope,  and  a  Bibliography 
to  date.  With  diagrams  and  folding  plates.  By 
Thomas  Nolan.  Second  edition,  revised  and  enlarged. 

No.  52.      IMAGINARY     QUANTITIES;       THEIR 

Geometrical  Interpretation.  Translated  from  the 
French  of  M.  Argand  by  Prof.  A.  S.  Hardy. 

No.  53.    INDUCTION  COILS ;    HOW  MADE  AND 

How  Used.     Eleventh  American  edition. 

No.  54.    KINEMATICS     OF     MACHINERY.      By 

Prof.  Alex.  B.  W.  Kennedy.  With  an  Introduction  by 
Prof.  R.  H.  Thurston. 

No.  55.    SEWER  GASES;    THEIR  NATURE  AND 

Origin.  By  A.  de  Varona.  Second  edition,  revised  and 
enlarged. 

*No.  56.  THE  ACTUAL  LATERAL  PRESSURE 

of  Earthwork.     By  Benj.  Baker,  M.  Inst.,  C.E. 

No.  57.      INCANDESCENT    ELECTRIC    LIGHT- 

ing.  A  Practical  Description  of  the  Edison  System. 
By  L.  H.  Latimer.  To  which  is  added  the  Design  and 
Operation  of  the  Incandescent  Stations,  by  C.  J.  Field; 
and  the  Maximum  Efficiency  of  Incandescent  Lamps, 
by  John  W.  Howell. 

No.  58.     VENTILATION   OF   COAL  MINES.     By 

W.  Fairley,  M.E.,  and  Geo.  J.  Andr6. 

No.  59.    RAILROAD  ECONOMICS;    OR,  NOTES 

With  Comments.     By  S.  W.  Robinson,  C.E. 

No.  60.     STRENGTH        OF       WROUGHT-IRON 

Bridge  Members.     By  S.  W.  Robinson,  C.E. 
No.  61.    POTABLE  WATER,  AND  METHODS  OF 

Detecting  Impurities.     By  M.  N.  Baker.     Second  edi- 
tion, revised  and  enlarged. 
No.  62.     THEORY    OF   THE    GAS-ENGINE.      By 

Dougald  Clerk.  Third  edition.  With  additional 
matter.  Edited  by  F.  E.  Idell,  M.E. 


THE   VAN   NQSTRAND   SCIENCE    SERIES 

No.  63.    HOUSE-DRAINAGE     AND     SANITARY 

Plumbing.     By  W.  P.  Gerhard.     Twelfth  edition. 

No.  64.  ELECTROMAGNETS.  By  A.  N.  Mans- 
field. Second  edition,  revised. 

No.  65.     POCKET     LOGARITHMS      TO     FOUR 

Places  of  Decimals.  Including  Logarithms  of  Num- 
bers, etc. 

No.  66.    DYNAMO-ELECTRIC  MACHINERY.   By 

S.  P.  Thompson.  With  an  Introduction  by  F.  L.  Pope. 
Third  edition,  revised. 

No.  67.     HYDRAULIC   TABLES  FOR  THE   CAL- 

culation  of  the  Discharge  through  Sewers,  Pipes,  and 
Conduits.  Based  on  "Kutter's  Formula."  By  P.  J. 
Flynn. 

No.  68.     STEAM-HEATING.     By   Robert   Bi 

Third  edition,  revised,  with  additions  by  A.  R.  Wol 

No.  69.    CHEMICAL   PROBLEMS.      By  Prof.  J.  C. 

Foye.     Fifth  edition,  revised  and  enlarged. 

No.  70.    EXPLOSIVE       MATERIALS.     By    Lieut. 

John  P.  Wisser. 

No.  71.     DYNAMIC       ELECTRICITY.     By    John 

Hopkinson,  J.  N.  Shoolbred,  and  R.  E.  Day. 
No.  72.    TOPOGRAPHICAL       SURVEYING.     By 

George  J.  Specht,  Prof.  A.  S.  Hardy,  John  B.  McMaster, 
and  H.  F.  Walling.  Fourth  edition,  revised. 

No.  73.  SYMBOLIC  ALGEBRA;  OR,  THE  ALGE- 
bra  of  Algebraic  Numbers.  By  Prof.  William  Cain. 

No.  74.  TESTING  MACHINES;  THEIR  HIS- 
tory,  Construction  and  Use.  By  Arthur  V.  Abbott. 

No.  75.  RECENT  PROGRESS  IN  DYNAMO- 
electric  Machines.  Being  a  Supplement  to  "Dynamo- 
electric  Machinery.  By  Prof.  Sylvanus  P.  Thompson. 

No.  76.  MODERN  REPRODUCTIVE  GRAPHIC 
Processes.  By  Lieut.  James  S.  Pettit,  U.S.A. 

No.  77.  STADIA  SURVEYING.  The  Theory  of 
Stadia  Measurements.  By  Arthur  Winslow.  Eighth 
edition. 

No.  78.    THE      STEAM-ENGINE      INDICATOR 

and  Its  Use.     By  W.  B.  Le  Van. 
No.  79.     THE    FIGURE    OF    THE    EARTH.     By 

Frank  C.  Roberts,  C.E. 
No.  8O.    HEALTHY          FOUNDATIONS          FOR 

Houses.     By  Glenn  Brown. 
*No.  81.    WATER     METERS:       COMPARATIVE 

Tests  of  Accuracy,  Delivery,  etc.  Distinctive  Features 
of  the  Worthing(,on,  Kennedy,  Siemens,  and  Hesse 
meters.  By  Ross  E.  Browne. 


DIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 


Return  to  desk  from  which  borrowed. 
is  book  is  DUE  on  the  last  date  stamped  below. 


MAY  3     1948 

IUMJO  1948 

1  1 

DEC  13  1' 


JUN     1  1949 


JAN  2  2  19I3C 


(A5702sl6)476 


YA  03C9G 


